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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sqrtcl 14101 | Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ) | ||
Theorem | sqrtthlem 14102 | Lemma for sqrtth 14104. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+)) | ||
Theorem | sqrtf 14103 | Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.) |
⊢ √:ℂ⟶ℂ | ||
Theorem | sqrtth 14104 | 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 14105 | 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 14106 | 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 14107 | 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 14108 | 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 14109 | Arithmetic-geometric mean inequality for 𝑛 = 2. (Contributed by Mario Carneiro, 2-Jul-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) ≤ ((𝐴 + 𝐵) / 2)) | ||
Theorem | sqrtthi 14110 | 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 14111 | 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 14112 | 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 14113 | Square root of square. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqi 14114 | Square root of square. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqsqrti 14115 | Square of square root. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴) | ||
Theorem | sqrtge0i 14116 | 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 14117 | A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴) | ||
Theorem | absnidi 14118 | 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 14119 | 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 14120 | 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 14121 | Absolute value of a real number. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (abs‘𝐴) = (√‘(𝐴↑2)) | ||
Theorem | sqrtpclii 14122 | 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 14123 | 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 14124 | 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 14125 | 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 14126 | 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 14127 | 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 14128 | Square root is monotonic. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
Theorem | sqrtlti 14129 | 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 14130 | 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 14131 | 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 | absvalsqi 14132 | Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)) | ||
Theorem | absvalsq2i 14133 | 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 14134 | Real closure of absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘𝐴) ∈ ℝ | ||
Theorem | absge0i 14135 | Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (abs‘𝐴) | ||
Theorem | absval2i 14136 | 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 14137 | 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 14138 | 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 14139 | Absolute value of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘-𝐴) = (abs‘𝐴) | ||
Theorem | abscji 14140 | 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 14141 | 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 14142 | 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 14143 | 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 14144 | 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 14145 | 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 14146 | 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 14147 | 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 14148 | 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 14149 | 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 14150 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) | ||
Theorem | sqrtgt0d 14151 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 < (√‘𝐴)) | ||
Theorem | absnidd 14152 | 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 14153 | 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 14154 | 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 14155 | Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2))) | ||
Theorem | resqrtcld 14156 | 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 14157 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqd 14158 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqrtge0d 14159 | 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 14160 | 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 14161 | 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 14162 | 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 14163 | 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 14164 | 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 14165 | 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 14166 | 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 14167 | 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 14168 | 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 14169 | 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 14170 | 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 14171 | 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 14172 | 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 14173 | 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 14174 | 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 14175 | Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) | ||
Theorem | sqrtcld 14176 | 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 14177 | 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 14178 | 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 14179 | 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 14180 | 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 14181 | 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 14182 | 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 14183 | Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) | ||
Theorem | absval2d 14184 | 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 14185 | 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 14186 | 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 14187 | 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 14188 | Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) | ||
Theorem | abscjd 14189 | 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 14190 | 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 14191 | 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 14192 | 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 14193 | 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 14194 | 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 14195 | 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 14196 | 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 14197 | 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 14198 | 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 14199 | 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 14200 | 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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