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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rexfiuz 10601* | Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.) |
⊢ (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) | ||
Theorem | rexuz3 10602* | Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) | ||
Theorem | rexanuz2 10603* | Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) | ||
Theorem | r19.29uz 10604* | A version of 19.29 1567 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((∀𝑘 ∈ 𝑍 𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)) | ||
Theorem | r19.2uz 10605* | A version of r19.2m 3396 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) | ||
Theorem | recvguniqlem 10606 | Lemma for recvguniq 10607. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.) |
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐴 < ((𝐹‘𝐾) + ((𝐴 − 𝐵) / 2))) & ⊢ (𝜑 → (𝐹‘𝐾) < (𝐵 + ((𝐴 − 𝐵) / 2))) ⇒ ⊢ (𝜑 → ⊥) | ||
Theorem | recvguniq 10607* | Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.) |
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) < (𝐿 + 𝑥) ∧ 𝐿 < ((𝐹‘𝑘) + 𝑥))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) < (𝑀 + 𝑥) ∧ 𝑀 < ((𝐹‘𝑘) + 𝑥))) ⇒ ⊢ (𝜑 → 𝐿 = 𝑀) | ||
Syntax | csqrt 10608 | Extend class notation to include square root of a complex number. |
class √ | ||
Syntax | cabs 10609 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
class abs | ||
Definition | df-rsqrt 10610* |
Define a function whose value is the square root of a nonnegative real
number.
Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root. (Contributed by Jim Kingdon, 23-Aug-2020.) |
⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦))) | ||
Definition | df-abs 10611 | Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.) |
⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | ||
Theorem | sqrtrval 10612* | Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.) |
⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) | ||
Theorem | absval 10613 | The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | ||
Theorem | rennim 10614 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) | ||
Theorem | sqrt0rlem 10615 | Lemma for sqrt0 10616. (Contributed by Jim Kingdon, 26-Aug-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ ((𝐴↑2) = 0 ∧ 0 ≤ 𝐴)) ↔ 𝐴 = 0) | ||
Theorem | sqrt0 10616 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ (√‘0) = 0 | ||
Theorem | resqrexlem1arp 10617 | Lemma for resqrex 10638. 1 + 𝐴 is a positive real (expressed in a way that will help apply seqf 10075 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+) | ||
Theorem | resqrexlemp1rp 10618* | Lemma for resqrex 10638. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10075 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ (𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵(𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝐶) ∈ ℝ+) | ||
Theorem | resqrexlemf 10619* | Lemma for resqrex 10638. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) | ||
Theorem | resqrexlemf1 10620* | Lemma for resqrex 10638. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘1) = (1 + 𝐴)) | ||
Theorem | resqrexlemfp1 10621* | Lemma for resqrex 10638. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2)) | ||
Theorem | resqrexlemover 10622* | Lemma for resqrex 10638. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁)↑2)) | ||
Theorem | resqrexlemdec 10623* | Lemma for resqrex 10638. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁)) | ||
Theorem | resqrexlemdecn 10624* | Lemma for resqrex 10638. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 < 𝑀) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁)) | ||
Theorem | resqrexlemlo 10625* | Lemma for resqrex 10638. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 / (2↑𝑁)) < (𝐹‘𝑁)) | ||
Theorem | resqrexlemcalc1 10626* | Lemma for resqrex 10638. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) = (((((𝐹‘𝑁)↑2) − 𝐴)↑2) / (4 · ((𝐹‘𝑁)↑2)))) | ||
Theorem | resqrexlemcalc2 10627* | Lemma for resqrex 10638. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) ≤ ((((𝐹‘𝑁)↑2) − 𝐴) / 4)) | ||
Theorem | resqrexlemcalc3 10628* | Lemma for resqrex 10638. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) | ||
Theorem | resqrexlemnmsq 10629* | Lemma for resqrex 10638. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ≤ 𝑀) ⇒ ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) | ||
Theorem | resqrexlemnm 10630* | Lemma for resqrex 10638. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝑁) − (𝐹‘𝑀)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑁 − 1)))) | ||
Theorem | resqrexlemcvg 10631* | Lemma for resqrex 10638. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹‘𝑖) + 𝑥))) | ||
Theorem | resqrexlemgt0 10632* | Lemma for resqrex 10638. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) ⇒ ⊢ (𝜑 → 0 ≤ 𝐿) | ||
Theorem | resqrexlemoverl 10633* | Lemma for resqrex 10638. Every term in the sequence is an overestimate compared with the limit 𝐿. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ (𝜑 → 𝐾 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐿 ≤ (𝐹‘𝐾)) | ||
Theorem | resqrexlemglsq 10634* | Lemma for resqrex 10638. The sequence formed by squaring each term of 𝐹 converges to (𝐿↑2). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) ⇒ ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐺‘𝑘) < ((𝐿↑2) + 𝑒) ∧ (𝐿↑2) < ((𝐺‘𝑘) + 𝑒))) | ||
Theorem | resqrexlemga 10635* | Lemma for resqrex 10638. The sequence formed by squaring each term of 𝐹 converges to 𝐴. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) ⇒ ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐺‘𝑘) < (𝐴 + 𝑒) ∧ 𝐴 < ((𝐺‘𝑘) + 𝑒))) | ||
Theorem | resqrexlemsqa 10636* | Lemma for resqrex 10638. The square of a limit is 𝐴. (Contributed by Jim Kingdon, 7-Aug-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) ⇒ ⊢ (𝜑 → (𝐿↑2) = 𝐴) | ||
Theorem | resqrexlemex 10637* | Lemma for resqrex 10638. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) | ||
Theorem | resqrex 10638* | Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴)) | ||
Theorem | rsqrmo 10639* | Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) | ||
Theorem | rersqreu 10640* | Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) | ||
Theorem | resqrtcl 10641 | Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) | ||
Theorem | rersqrtthlem 10642 | Lemma for resqrtth 10643. (Contributed by Jim Kingdon, 10-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴))) | ||
Theorem | resqrtth 10643 | Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴) | ||
Theorem | remsqsqrt 10644 | Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) · (√‘𝐴)) = 𝐴) | ||
Theorem | sqrtge0 10645 | The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (√‘𝐴)) | ||
Theorem | sqrtgt0 10646 | The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴)) | ||
Theorem | sqrtmul 10647 | Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵))) | ||
Theorem | sqrtle 10648 | Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵))) | ||
Theorem | sqrtlt 10649 | Square root is strictly monotonic. Closed form of sqrtlti 10749. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵))) | ||
Theorem | sqrt11ap 10650 | Analogue to sqrt11 10651 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) # (√‘𝐵) ↔ 𝐴 # 𝐵)) | ||
Theorem | sqrt11 10651 | The square root function is one-to-one. Also see sqrt11ap 10650 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | sqrt00 10652 | A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | rpsqrtcl 10653 | The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.) |
⊢ (𝐴 ∈ ℝ+ → (√‘𝐴) ∈ ℝ+) | ||
Theorem | sqrtdiv 10654 | Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵))) | ||
Theorem | sqrtsq2 10655 | Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) = 𝐵 ↔ 𝐴 = (𝐵↑2))) | ||
Theorem | sqrtsq 10656 | Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴↑2)) = 𝐴) | ||
Theorem | sqrtmsq 10657 | Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴 · 𝐴)) = 𝐴) | ||
Theorem | sqrt1 10658 | The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.) |
⊢ (√‘1) = 1 | ||
Theorem | sqrt4 10659 | The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.) |
⊢ (√‘4) = 2 | ||
Theorem | sqrt9 10660 | The square root of 9 is 3. (Contributed by NM, 11-May-2004.) |
⊢ (√‘9) = 3 | ||
Theorem | sqrt2gt1lt2 10661 | The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.) |
⊢ (1 < (√‘2) ∧ (√‘2) < 2) | ||
Theorem | absneg 10662 | Absolute value of negative. (Contributed by NM, 27-Feb-2005.) |
⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | ||
Theorem | abscl 10663 | Real closure of absolute value. (Contributed by NM, 3-Oct-1999.) |
⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | ||
Theorem | abscj 10664 | The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (abs‘𝐴)) | ||
Theorem | absvalsq 10665 | Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | ||
Theorem | absvalsq2 10666 | Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.) |
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))) | ||
Theorem | sqabsadd 10667 | Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) | ||
Theorem | sqabssub 10668 | Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) | ||
Theorem | absval2 10669 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.) |
⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))) | ||
Theorem | abs0 10670 | The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ (abs‘0) = 0 | ||
Theorem | absi 10671 | The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
⊢ (abs‘i) = 1 | ||
Theorem | absge0 10672 | Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | ||
Theorem | absrpclap 10673 | The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ+) | ||
Theorem | abs00ap 10674 | The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.) |
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0)) | ||
Theorem | absext 10675 | Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) # (abs‘𝐵) → 𝐴 # 𝐵)) | ||
Theorem | abs00 10676 | The absolute value of a number is zero iff the number is zero. Also see abs00ap 10674 which is similar but for apartness. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | abs00ad 10677 | A complex number is zero iff its absolute value is zero. Deduction form of abs00 10676. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | abs00bd 10678 | If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 = 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) = 0) | ||
Theorem | absreimsq 10679 | Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴↑2) + (𝐵↑2))) | ||
Theorem | absreim 10680 | Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 + (i · 𝐵))) = (√‘((𝐴↑2) + (𝐵↑2)))) | ||
Theorem | absmul 10681 | Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) | ||
Theorem | absdivap 10682 | Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | ||
Theorem | absid 10683 | A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) | ||
Theorem | abs1 10684 | The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
⊢ (abs‘1) = 1 | ||
Theorem | absnid 10685 | A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (abs‘𝐴) = -𝐴) | ||
Theorem | leabs 10686 | A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴)) | ||
Theorem | qabsor 10687 | The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.) |
⊢ (𝐴 ∈ ℚ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | ||
Theorem | qabsord 10688 | The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) | ||
Theorem | absre 10689 | Absolute value of a real number. (Contributed by NM, 17-Mar-2005.) |
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴↑2))) | ||
Theorem | absresq 10690 | Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.) |
⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) | ||
Theorem | absexp 10691 | Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)) | ||
Theorem | absexpzap 10692 | Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)) | ||
Theorem | abssq 10693 | Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2))) | ||
Theorem | sqabs 10694 | The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵))) | ||
Theorem | absrele 10695 | The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.) |
⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘𝐴)) ≤ (abs‘𝐴)) | ||
Theorem | absimle 10696 | The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) | ||
Theorem | nn0abscl 10697 | The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) |
⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0) | ||
Theorem | zabscl 10698 | The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℤ) | ||
Theorem | ltabs 10699 | A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < (abs‘𝐴)) → 𝐴 < 0) | ||
Theorem | abslt 10700 | Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴 ∧ 𝐴 < 𝐵))) |
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