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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | lttri3d 11401 | Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
| Theorem | lttri4d 11402 | Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | ||
| Theorem | letri3d 11403 | Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
| Theorem | leloed 11404 | 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | ||
| Theorem | eqleltd 11405 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
| Theorem | ltlend 11406 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) | ||
| Theorem | lenltd 11407 | 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
| Theorem | ltnled 11408 | 'Less than' in terms 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 | ltled 11409 | 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
| Theorem | ltnsymd 11410 | 'Less than' implies '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 | nltled 11411 | 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ¬ 𝐵 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
| Theorem | lensymd 11412 | 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | ||
| Theorem | letrid 11413 | Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
| Theorem | leltned 11414 | 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | ||
| Theorem | leneltd 11415 | 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
| Theorem | mulgt0d 11416 | The product of two 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 | ltadd2d 11417 | Addition to 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 + 𝐴) < (𝐶 + 𝐵))) | ||
| Theorem | letrd 11418 | Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
| Theorem | lelttrd 11419 | Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
| Theorem | ltadd2dd 11420 | Addition to 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 + 𝐴) < (𝐶 + 𝐵)) | ||
| Theorem | ltletrd 11421 | Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
| Theorem | lttrd 11422 | Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
| Theorem | lelttrdi 11423 | 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)) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) | ||
| Theorem | dedekind 11424* | The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 11233 with appropriate adjustments, states that, if 𝐴 completely preceeds 𝐵, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.) |
| ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | ||
| Theorem | dedekindle 11425* | The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.) |
| ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | ||
| Theorem | mul12 11426 | Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | ||
| Theorem | mul32 11427 | Commutative/associative law. (Contributed by NM, 8-Oct-1999.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | ||
| Theorem | mul31 11428 | Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) | ||
| Theorem | mul4 11429 | Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | ||
| Theorem | mul4r 11430 | Rearrangement of 4 factors: swap the right factors in the factors of a product of two products. (Contributed by AV, 4-Mar-2023.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐷) · (𝐶 · 𝐵))) | ||
| Theorem | muladd11 11431 | A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) | ||
| Theorem | 1p1times 11432 | Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | ||
| Theorem | peano2cn 11433 | A theorem for complex numbers analogous the second Peano postulate peano2nn 12278. (Contributed by NM, 17-Aug-2005.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ) | ||
| Theorem | peano2re 11434 | A theorem for reals analogous the second Peano postulate peano2nn 12278. (Contributed by NM, 5-Jul-2005.) |
| ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | ||
| Theorem | readdcan 11435 | Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | 00id 11436 | 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ (0 + 0) = 0 | ||
| Theorem | mul02lem1 11437 | Lemma for mul02 11439. If any real does not produce 0 when multiplied by 0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 𝐵 ∈ ℂ) → 𝐵 = (𝐵 + 𝐵)) | ||
| Theorem | mul02lem2 11438 | Lemma for mul02 11439. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) | ||
| Theorem | mul02 11439 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | ||
| Theorem | mul01 11440 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | ||
| Theorem | addrid 11441 | 0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | ||
| Theorem | cnegex 11442* | Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) | ||
| Theorem | cnegex2 11443* | Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) | ||
| Theorem | addlid 11444 | 0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | ||
| Theorem | addcan 11445 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) | ||
| Theorem | addcan2 11446 | Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) | ||
| Theorem | addcom 11447 | Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
| Theorem | addridi 11448 | 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 + 0) = 𝐴 | ||
| Theorem | addlidi 11449 | 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (0 + 𝐴) = 𝐴 | ||
| Theorem | mul02i 11450 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (0 · 𝐴) = 0 | ||
| Theorem | mul01i 11451 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 0) = 0 | ||
| Theorem | addcomi 11452 | Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) | ||
| Theorem | addcomli 11453 | Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ (𝐵 + 𝐴) = 𝐶 | ||
| Theorem | addcani 11454 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶) | ||
| Theorem | addcan2i 11455 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵) | ||
| Theorem | mul12i 11456 | Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)) | ||
| Theorem | mul32i 11457 | Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵) | ||
| Theorem | mul4i 11458 | Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)) | ||
| Theorem | mul02d 11459 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (0 · 𝐴) = 0) | ||
| Theorem | mul01d 11460 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 0) = 0) | ||
| Theorem | addridd 11461 | 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐴) | ||
| Theorem | addlidd 11462 | 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐴) | ||
| Theorem | addcomd 11463 | Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
| Theorem | addcand 11464 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) | ||
| Theorem | addcan2d 11465 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) | ||
| Theorem | addcanad 11466 | Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 11464. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
| Theorem | addcan2ad 11467 | Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 11465. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | addneintrd 11468 | Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 11466. Consequence of addcand 11464. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶)) | ||
| Theorem | addneintr2d 11469 | Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 11467. Consequence of addcan2d 11465. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) | ||
| Theorem | mul12d 11470 | Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | ||
| Theorem | mul32d 11471 | Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | ||
| Theorem | mul31d 11472 | Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) | ||
| Theorem | mul4d 11473 | Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | ||
| Theorem | muladd11r 11474 | A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1)) | ||
| Theorem | comraddd 11475 | Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
| ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) | ||
| Theorem | comraddi 11476 | Commute RHS addition. See addcomli 11453 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
| ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ 𝐴 = (𝐶 + 𝐵) | ||
| Theorem | ltaddneg 11477 | 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)) | ||
| Theorem | ltaddnegr 11478 | Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵)) | ||
| Theorem | add12 11479 | Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) | ||
| Theorem | add32 11480 | Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)) | ||
| Theorem | add32r 11481 | Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = ((𝐴 + 𝐶) + 𝐵)) | ||
| Theorem | add4 11482 | Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | ||
| Theorem | add42 11483 | Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) | ||
| Theorem | add12i 11484 | Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)) | ||
| Theorem | add32i 11485 | Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵) | ||
| Theorem | add4i 11486 | Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)) | ||
| Theorem | add42i 11487 | Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) (Proof shortened by OpenAI, 25-Mar-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)) | ||
| Theorem | add12d 11488 | Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) | ||
| Theorem | add32d 11489 | Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)) | ||
| Theorem | add4d 11490 | Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | ||
| Theorem | add42d 11491 | Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))) | ||
| Syntax | cmin 11492 | Extend class notation to include subtraction. |
| class − | ||
| Syntax | cneg 11493 | Extend class notation to include unary minus. The symbol - is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-) and subtraction cmin 11492 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 7431, if we used the same symbol then "( − 𝐴 − 𝐵) " could mean either "− 𝐴 " minus "𝐵", or it could represent the (meaningless) operation of classes "− " and "− 𝐵 " connected with "operation" "𝐴". On the other hand, "(-𝐴 − 𝐵) " is unambiguous. |
| class -𝐴 | ||
| Definition | df-sub 11494* | Define subtraction. Theorem subval 11499 shows its value (and describes how this definition works), Theorem subaddi 11596 relates it to addition, and Theorems subcli 11585 and resubcli 11571 prove its closure laws. (Contributed by NM, 26-Nov-1994.) |
| ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) | ||
| Definition | df-neg 11495 | Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction (−) to prevent syntax ambiguity. See cneg 11493 for a discussion of this. (Contributed by NM, 10-Feb-1995.) |
| ⊢ -𝐴 = (0 − 𝐴) | ||
| Theorem | 0cnALT 11496 | Alternate proof of 0cn 11253 which does not reference ax-1cn 11213. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 0 ∈ ℂ | ||
| Theorem | 0cnALT2 11497 | Alternate proof of 0cnALT 11496 which is shorter, but depends on ax-8 2110, ax-13 2377, ax-sep 5296, ax-nul 5306, ax-pow 5365, ax-pr 5432, ax-un 7755, and every complex number axiom except ax-pre-mulgt0 11232 and ax-pre-sup 11233. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 0 ∈ ℂ | ||
| Theorem | negeu 11498* | Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) | ||
| Theorem | subval 11499* | Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) | ||
| Theorem | negeq 11500 | Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.) |
| ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | ||
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