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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | sqrtthlem 15401 | Lemma for sqrtth 15403. (Contributed by Mario Carneiro, 10-Jul-2013.) |
| ⊢ (𝐴 ∈ ℂ → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+)) | ||
| Theorem | sqrtf 15402 | Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ √:ℂ⟶ℂ | ||
| Theorem | sqrtth 15403 | Square root theorem over the complex numbers. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.) |
| ⊢ (𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴) | ||
| Theorem | sqrtrege0 15404 | The square root function must make a choice between the two roots, which differ by a sign change. In the general complex case, the choice of "positive" and "negative" is not so clear. The convention we use is to take the root with positive real part, unless 𝐴 is a nonpositive real (in which case both roots have 0 real part); in this case we take the one in the positive imaginary direction. Another way to look at this is that we choose the root that is largest with respect to lexicographic order on the complex numbers (sorting by real part first, then by imaginary part as tie-breaker). (Contributed by Mario Carneiro, 10-Jul-2013.) |
| ⊢ (𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴))) | ||
| Theorem | eqsqrtor 15405 | Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = 𝐵 ↔ (𝐴 = (√‘𝐵) ∨ 𝐴 = -(√‘𝐵)))) | ||
| Theorem | eqsqrtd 15406 | A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴↑2) = 𝐵) & ⊢ (𝜑 → 0 ≤ (ℜ‘𝐴)) & ⊢ (𝜑 → ¬ (i · 𝐴) ∈ ℝ+) ⇒ ⊢ (𝜑 → 𝐴 = (√‘𝐵)) | ||
| Theorem | eqsqrt2d 15407 | A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴↑2) = 𝐵) & ⊢ (𝜑 → 0 < (ℜ‘𝐴)) ⇒ ⊢ (𝜑 → 𝐴 = (√‘𝐵)) | ||
| Theorem | amgm2 15408 | Arithmetic-geometric mean inequality for 𝑛 = 2. (Contributed by Mario Carneiro, 2-Jul-2014.) (Proof shortened by AV, 9-Jul-2022.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) ≤ ((𝐴 + 𝐵) / 2)) | ||
| Theorem | sqrtthi 15409 | Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → ((√‘𝐴) · (√‘𝐴)) = 𝐴) | ||
| Theorem | sqrtcli 15410 | The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘𝐴) ∈ ℝ) | ||
| Theorem | sqrtgt0i 15411 | The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 < 𝐴 → 0 < (√‘𝐴)) | ||
| Theorem | sqrtmsqi 15412 | Square root of square. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
| Theorem | sqrtsqi 15413 | Square root of square. (Contributed by NM, 11-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴) | ||
| Theorem | sqsqrti 15414 | Square of square root. (Contributed by NM, 11-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴) | ||
| Theorem | sqrtge0i 15415 | The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → 0 ≤ (√‘𝐴)) | ||
| Theorem | absidi 15416 | A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴) | ||
| Theorem | absnidi 15417 | A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴) | ||
| Theorem | leabsi 15418 | A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ≤ (abs‘𝐴) | ||
| Theorem | absori 15419 | The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴) | ||
| Theorem | absrei 15420 | Absolute value of a real number. (Contributed by NM, 3-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (abs‘𝐴) = (√‘(𝐴↑2)) | ||
| Theorem | sqrtpclii 15421 | The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ (√‘𝐴) ∈ ℝ | ||
| Theorem | sqrtgt0ii 15422 | The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 0 < (√‘𝐴) | ||
| Theorem | sqrt11i 15423 | The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | sqrtmuli 15424 | Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))) | ||
| Theorem | sqrtmulii 15425 | Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 ≤ 𝐴 & ⊢ 0 ≤ 𝐵 ⇒ ⊢ (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)) | ||
| Theorem | sqrtmsq2i 15426 | Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐵))) | ||
| Theorem | sqrtlei 15427 | Square root is monotonic. (Contributed by NM, 3-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
| Theorem | sqrtlti 15428 | Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵))) | ||
| Theorem | abslti 15429 | Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵)) | ||
| Theorem | abslei 15430 | Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) | ||
| Theorem | cnsqrt00 15431 | A square root of a complex number is zero iff its argument is 0. Version of sqrt00 15302 for complex numbers. (Contributed by AV, 26-Jan-2023.) |
| ⊢ (𝐴 ∈ ℂ → ((√‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
| Theorem | absvalsqi 15432 | Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)) | ||
| Theorem | absvalsq2i 15433 | Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)) | ||
| Theorem | abscli 15434 | Real closure of absolute value. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘𝐴) ∈ ℝ | ||
| Theorem | absge0i 15435 | Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (abs‘𝐴) | ||
| Theorem | absval2i 15436 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))) | ||
| Theorem | abs00i 15437 | The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) = 0 ↔ 𝐴 = 0) | ||
| Theorem | absgt0i 15438 | The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)) | ||
| Theorem | absnegi 15439 | Absolute value of negative. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘-𝐴) = (abs‘𝐴) | ||
| Theorem | abscji 15440 | The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘(∗‘𝐴)) = (abs‘𝐴) | ||
| Theorem | releabsi 15441 | The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (ℜ‘𝐴) ≤ (abs‘𝐴) | ||
| Theorem | abssubi 15442 | Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)) | ||
| Theorem | absmuli 15443 | Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)) | ||
| Theorem | sqabsaddi 15444 | Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) | ||
| Theorem | sqabssubi 15445 | Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) | ||
| Theorem | absdivzi 15446 | Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | ||
| Theorem | abstrii 15447 | Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)) | ||
| Theorem | abs3difi 15448 | Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))) | ||
| Theorem | abs3lemi 15449 | Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ (((abs‘(𝐴 − 𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) → (abs‘(𝐴 − 𝐵)) < 𝐷) | ||
| Theorem | rpsqrtcld 15450 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) | ||
| Theorem | sqrtgt0d 15451 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 < (√‘𝐴)) | ||
| Theorem | absnidd 15452 | A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) = -𝐴) | ||
| Theorem | leabsd 15453 | A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ (abs‘𝐴)) | ||
| Theorem | absord 15454 | The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | ||
| Theorem | absred 15455 | Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2))) | ||
| Theorem | resqrtcld 15456 | The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ) | ||
| Theorem | sqrtmsqd 15457 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
| Theorem | sqrtsqd 15458 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴) | ||
| Theorem | sqrtge0d 15459 | The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ (√‘𝐴)) | ||
| Theorem | sqrtnegd 15460 | The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘-𝐴) = (i · (√‘𝐴))) | ||
| Theorem | absidd 15461 | A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (abs‘𝐴) = 𝐴) | ||
| Theorem | sqrtdivd 15462 | Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵))) | ||
| Theorem | sqrtmuld 15463 | Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))) | ||
| Theorem | sqrtsq2d 15464 | Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵↑2))) | ||
| Theorem | sqrtled 15465 | Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
| Theorem | sqrtltd 15466 | Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵))) | ||
| Theorem | sqr11d 15467 | The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → (√‘𝐴) = (√‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | absltd 15468 | Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) | ||
| Theorem | absled 15469 | Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | ||
| Theorem | abssubge0d 15470 | Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (𝐵 − 𝐴)) | ||
| Theorem | abssuble0d 15471 | Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) | ||
| Theorem | absdifltd 15472 | The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐶 ↔ ((𝐵 − 𝐶) < 𝐴 ∧ 𝐴 < (𝐵 + 𝐶)))) | ||
| Theorem | absdifled 15473 | The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) ≤ 𝐶 ↔ ((𝐵 − 𝐶) ≤ 𝐴 ∧ 𝐴 ≤ (𝐵 + 𝐶)))) | ||
| Theorem | icodiamlt 15474 | Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵))) → (abs‘(𝐶 − 𝐷)) < (𝐵 − 𝐴)) | ||
| Theorem | abscld 15475 | Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) | ||
| Theorem | sqrtcld 15476 | Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℂ) | ||
| Theorem | sqrtrege0d 15477 | The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (ℜ‘(√‘𝐴))) | ||
| Theorem | sqsqrtd 15478 | Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((√‘𝐴)↑2) = 𝐴) | ||
| Theorem | msqsqrtd 15479 | Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((√‘𝐴) · (√‘𝐴)) = 𝐴) | ||
| Theorem | sqr00d 15480 | A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (√‘𝐴) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | absvalsqd 15481 | Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | ||
| Theorem | absvalsq2d 15482 | Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))) | ||
| Theorem | absge0d 15483 | Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) | ||
| Theorem | absval2d 15484 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))) | ||
| Theorem | abs00d 15485 | The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | absne0d 15486 | The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) ≠ 0) | ||
| Theorem | absrpcld 15487 | The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) | ||
| Theorem | absnegd 15488 | Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) | ||
| Theorem | abscjd 15489 | The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(∗‘𝐴)) = (abs‘𝐴)) | ||
| Theorem | releabsd 15490 | The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℜ‘𝐴) ≤ (abs‘𝐴)) | ||
| Theorem | absexpd 15491 | Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)) | ||
| Theorem | abssubd 15492 | Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) | ||
| Theorem | absmuld 15493 | Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) | ||
| Theorem | absdivd 15494 | Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | ||
| Theorem | abstrid 15495 | Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) | ||
| Theorem | abs2difd 15496 | Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))) | ||
| Theorem | abs2dif2d 15497 | Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵))) | ||
| Theorem | abs2difabsd 15498 | Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) | ||
| Theorem | abs3difd 15499 | Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵)))) | ||
| Theorem | abs3lemd 15500 | Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < (𝐷 / 2)) & ⊢ (𝜑 → (abs‘(𝐶 − 𝐵)) < (𝐷 / 2)) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝐷) | ||
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