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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eqsqrtor 15401 | 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 15402 | 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 15403 | 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 15404 | 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 15405 | 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 15406 | 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 15407 | 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 15408 | Square root of square. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqi 15409 | Square root of square. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqsqrti 15410 | Square of square root. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴) | ||
Theorem | sqrtge0i 15411 | 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 15412 | A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴) | ||
Theorem | absnidi 15413 | 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 15414 | 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 15415 | 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 15416 | Absolute value of a real number. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (abs‘𝐴) = (√‘(𝐴↑2)) | ||
Theorem | sqrtpclii 15417 | 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 15418 | 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 15419 | 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 15420 | 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 15421 | 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 15422 | 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 15423 | Square root is monotonic. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
Theorem | sqrtlti 15424 | 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 15425 | 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 15426 | 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 15427 | A square root of a complex number is zero iff its argument is 0. Version of sqrt00 15298 for complex numbers. (Contributed by AV, 26-Jan-2023.) |
⊢ (𝐴 ∈ ℂ → ((√‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | absvalsqi 15428 | Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)) | ||
Theorem | absvalsq2i 15429 | 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 15430 | Real closure of absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘𝐴) ∈ ℝ | ||
Theorem | absge0i 15431 | Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (abs‘𝐴) | ||
Theorem | absval2i 15432 | 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 15433 | 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 15434 | 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 15435 | Absolute value of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘-𝐴) = (abs‘𝐴) | ||
Theorem | abscji 15436 | 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 15437 | 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 15438 | 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 15439 | 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 15440 | 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 15441 | 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 15442 | 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 15443 | 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 15444 | 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 15445 | 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 15446 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) | ||
Theorem | sqrtgt0d 15447 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 < (√‘𝐴)) | ||
Theorem | absnidd 15448 | 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 15449 | 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 15450 | 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 15451 | Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2))) | ||
Theorem | resqrtcld 15452 | 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 15453 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqd 15454 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqrtge0d 15455 | 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 15456 | 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 15457 | 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 15458 | 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 15459 | 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 15460 | 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 15461 | 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 15462 | 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 15463 | 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 15464 | 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 15465 | 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 15466 | 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 15467 | 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 15468 | 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 15469 | 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 15470 | 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 15471 | Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) | ||
Theorem | sqrtcld 15472 | 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 15473 | 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 15474 | 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 15475 | 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 15476 | 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 15477 | 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 15478 | 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 15479 | Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) | ||
Theorem | absval2d 15480 | 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 15481 | 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 15482 | 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 15483 | 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 15484 | Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) | ||
Theorem | abscjd 15485 | 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 15486 | 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 15487 | 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 15488 | 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 15489 | 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 15490 | 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 15491 | 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 15492 | 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 15493 | 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 15494 | 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 15495 | 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 15496 | 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) | ||
Theorem | reusq0 15497* | A complex number is the square of exactly one complex number iff the given complex number is zero. (Contributed by AV, 21-Jun-2023.) |
⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0)) | ||
Theorem | bhmafibid1cn 15498 | The Brahmagupta-Fibonacci identity for complex numbers. Express the product of two sums of two squares as a sum of two squares. First result. (Contributed by Thierry Arnoux, 1-Feb-2020.) Generalization for complex numbers proposed by GL. (Revised by AV, 8-Jun-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2))) | ||
Theorem | bhmafibid2cn 15499 | The Brahmagupta-Fibonacci identity for complex numbers. Express the product of two sums of two squares as a sum of two squares. Second result. (Contributed by Thierry Arnoux, 1-Feb-2020.) Generalization for complex numbers proposed by GL. (Revised by AV, 8-Jun-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) + (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) − (𝐵 · 𝐶))↑2))) | ||
Theorem | bhmafibid1 15500 | The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. First result. Remark: The proof uses a different approach than the proof of bhmafibid1cn 15498, and is a little bit shorter. (Contributed by Thierry Arnoux, 1-Feb-2020.) (Proof modification is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2))) |
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