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Type | Label | Description |
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Statement | ||
Theorem | r19.2uz 14701* | A version of r19.2z 4438 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) | ||
Theorem | rexuzre 14702* | Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑗 ≤ 𝑘 → 𝜑))) | ||
Theorem | rexico 14703* | Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → 𝜑))) | ||
Theorem | cau3lem 14704* | Lemma for cau3 14705. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.) |
⊢ 𝑍 ⊆ ℤ & ⊢ (𝜏 → 𝜓) & ⊢ ((𝐹‘𝑘) = (𝐹‘𝑗) → (𝜓 ↔ 𝜒)) & ⊢ ((𝐹‘𝑘) = (𝐹‘𝑚) → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑘))) = (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗)))) & ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜒) → (𝐺‘((𝐹‘𝑚)𝐷(𝐹‘𝑗))) = (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑚)))) & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝑥 ∈ ℝ)) → (((𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ (𝐺‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) → (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜏 ∧ (𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜏 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(𝐺‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥))) | ||
Theorem | cau3 14705* | Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of 𝑗 in the assertion, so it can be used with rexanuz 14695 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)) | ||
Theorem | cau4 14706* | Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) ⇒ ⊢ (𝑁 ∈ 𝑍 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑊 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))) | ||
Theorem | caubnd2 14707* | A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦) | ||
Theorem | caubnd 14708* | A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦) | ||
Theorem | sqreulem 14709 | Lemma for sqreu 14710: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ 𝐵 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((𝐵↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝐵) ∧ (i · 𝐵) ∉ ℝ+)) | ||
Theorem | sqreu 14710* | Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) | ||
Theorem | sqrtcl 14711 | Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ) | ||
Theorem | sqrtthlem 14712 | Lemma for sqrtth 14714. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+)) | ||
Theorem | sqrtf 14713 | Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.) |
⊢ √:ℂ⟶ℂ | ||
Theorem | sqrtth 14714 | 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 14715 | 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 14716 | 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 14717 | 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 14718 | 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 14719 | 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 14720 | 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 14721 | 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 14722 | 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 14723 | Square root of square. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqi 14724 | Square root of square. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqsqrti 14725 | Square of square root. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴) | ||
Theorem | sqrtge0i 14726 | 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 14727 | A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴) | ||
Theorem | absnidi 14728 | 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 14729 | 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 14730 | 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 14731 | Absolute value of a real number. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (abs‘𝐴) = (√‘(𝐴↑2)) | ||
Theorem | sqrtpclii 14732 | 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 14733 | 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 14734 | 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 14735 | 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 14736 | 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 14737 | 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 14738 | Square root is monotonic. (Contributed by NM, 3-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
Theorem | sqrtlti 14739 | 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 14740 | 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 14741 | 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 14742 | A square root of a complex number is zero iff its argument is 0. Version of sqrt00 14613 for complex numbers. (Contributed by AV, 26-Jan-2023.) |
⊢ (𝐴 ∈ ℂ → ((√‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | absvalsqi 14743 | Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)) | ||
Theorem | absvalsq2i 14744 | 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 14745 | Real closure of absolute value. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘𝐴) ∈ ℝ | ||
Theorem | absge0i 14746 | Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ 0 ≤ (abs‘𝐴) | ||
Theorem | absval2i 14747 | 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 14748 | 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 14749 | 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 14750 | Absolute value of negative. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (abs‘-𝐴) = (abs‘𝐴) | ||
Theorem | abscji 14751 | 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 14752 | 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 14753 | 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 14754 | 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 14755 | 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 14756 | 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 14757 | 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 14758 | 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 14759 | 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 14760 | 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 14761 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (√‘𝐴) ∈ ℝ+) | ||
Theorem | sqrtgt0d 14762 | The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → 0 < (√‘𝐴)) | ||
Theorem | absnidd 14763 | 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 14764 | 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 14765 | 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 14766 | Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘𝐴) = (√‘(𝐴↑2))) | ||
Theorem | resqrtcld 14767 | 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 14768 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrtsqd 14769 | Square root of square. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqrtge0d 14770 | 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 14771 | 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 14772 | 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 14773 | 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 14774 | 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 14775 | 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 14776 | 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 14777 | 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 14778 | 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 14779 | 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 14780 | 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 14781 | 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 14782 | 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 14783 | 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 14784 | 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 14785 | 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 14786 | Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ) | ||
Theorem | sqrtcld 14787 | 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 14788 | 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 14789 | 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 14790 | 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 14791 | 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 14792 | 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 14793 | 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 14794 | Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → 0 ≤ (abs‘𝐴)) | ||
Theorem | absval2d 14795 | 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 14796 | 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 14797 | 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 14798 | 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 14799 | Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) | ||
Theorem | abscjd 14800 | 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)) |
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