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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addgt0d 8201 | 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 8202 | 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 | ltnegd 8203 | 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 8204 | 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 8205 | 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 8206 | 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 8207 | 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 8208 | 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 8209 | 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 8210 | 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 8211 | 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 8212 | 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 8213 | 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 8214 | 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 8215 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | suble0d 8216 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) | ||
Theorem | subge02d 8217 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴)) | ||
Theorem | ltadd1d 8218 | 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 8219 | 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 8220 | 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 8221 | '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 8222 | '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 8223 | '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 8224 | '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 8225 | '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 8226 | '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 8227 | '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 8228 | 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 8229 | 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 8230 | '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 8231 | '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 8232 | 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 8233 | 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 8234 | 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 8235 | 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 8236 | 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 8237 | 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 8238 | 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 8239 | 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 8240 | 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 8241 | 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 8242 | 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 | le2addd 8243 | 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 8244 | 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 8245 | 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 8246 | 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 8247 | 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 8248 | 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 8249 | 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 8250 | 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 8251 | 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 8252 | 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
⊢ 1 ≤ 1 | ||
Theorem | gt0add 8253 | A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < (𝐴 + 𝐵)) → (0 < 𝐴 ∨ 0 < 𝐵)) | ||
Syntax | creap 8254 | Class of real apartness relation. |
class #ℝ | ||
Definition | df-reap 8255* | Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 8262 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 8267). (Contributed by Jim Kingdon, 26-Jan-2020.) |
⊢ #ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))} | ||
Theorem | reapval 8256 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8268 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | reapirr 8257 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8285 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴) | ||
Theorem | recexre 8258* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | ||
Theorem | reapti 8259 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8302. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 #ℝ 𝐵)) | ||
Theorem | recexgt0 8260* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)) | ||
Syntax | cap 8261 | Class of complex apartness relation. |
class # | ||
Definition | df-ap 8262* |
Define complex apartness. Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8352 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8285), symmetry (apsym 8286), and cotransitivity (apcotr 8287). Apartness implies negated equality, as seen at apne 8303, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8302). (Contributed by Jim Kingdon, 26-Jan-2020.) |
⊢ # = {〈𝑥, 𝑦〉 ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))} | ||
Theorem | ixi 8263 | i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (i · i) = -1 | ||
Theorem | inelr 8264 | The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.) |
⊢ ¬ i ∈ ℝ | ||
Theorem | rimul 8265 | A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0) | ||
Theorem | rereim 8266 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 = (𝐵 + (i · 𝐶)))) → (𝐵 = 𝐴 ∧ 𝐶 = 0)) | ||
Theorem | apreap 8267 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ 𝐴 #ℝ 𝐵)) | ||
Theorem | reaplt 8268 | Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | reapltxor 8269 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ⊻ 𝐵 < 𝐴))) | ||
Theorem | 1ap0 8270 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
⊢ 1 # 0 | ||
Theorem | ltmul1a 8271 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) ∧ 𝐴 < 𝐵) → (𝐴 · 𝐶) < (𝐵 · 𝐶)) | ||
Theorem | ltmul1 8272 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) | ||
Theorem | lemul1 8273 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) | ||
Theorem | reapmul1lem 8274 | Lemma for reapmul1 8275. (Contributed by Jim Kingdon, 8-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶))) | ||
Theorem | reapmul1 8275 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8461. (Contributed by Jim Kingdon, 8-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶))) | ||
Theorem | reapadd1 8276 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶))) | ||
Theorem | reapneg 8277 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵)) | ||
Theorem | reapcotr 8278 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶))) | ||
Theorem | remulext1 8279 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵)) | ||
Theorem | remulext2 8280 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵)) | ||
Theorem | apsqgt0 8281 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴 · 𝐴)) | ||
Theorem | cru 8282 | The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | apreim 8283 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (𝐶 + (i · 𝐷)) ↔ (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) | ||
Theorem | mulreim 8284 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) · (𝐶 + (i · 𝐷))) = (((𝐴 · 𝐶) + -(𝐵 · 𝐷)) + (i · ((𝐶 · 𝐵) + (𝐷 · 𝐴))))) | ||
Theorem | apirr 8285 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
⊢ (𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴) | ||
Theorem | apsym 8286 | Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ 𝐵 # 𝐴)) | ||
Theorem | apcotr 8287 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶))) | ||
Theorem | apadd1 8288 | Addition respects apartness. Analogue of addcan 7865 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶))) | ||
Theorem | apadd2 8289 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵))) | ||
Theorem | addext 8290 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5737. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) | ||
Theorem | apneg 8291 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵)) | ||
Theorem | mulext1 8292 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵)) | ||
Theorem | mulext2 8293 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵)) | ||
Theorem | mulext 8294 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5737. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) | ||
Theorem | mulap0r 8295 | A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) # 0) → (𝐴 # 0 ∧ 𝐵 # 0)) | ||
Theorem | msqge0 8296 | A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) | ||
Theorem | msqge0i 8297 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 0 ≤ (𝐴 · 𝐴) | ||
Theorem | msqge0d 8298 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴)) | ||
Theorem | mulge0 8299 | 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)) | ||
Theorem | mulge0i 8300 | 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)) |
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