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Type | Label | Description |
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Statement | ||
Theorem | imaddd 11001 | Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | resubd 11002 | Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imsubd 11003 | Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remuld 11004 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immuld 11005 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjaddd 11006 | Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmuld 11007 | Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | ipcnd 11008 | Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjdivapd 11009 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | rered 11010 | A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | reim0d 11011 | The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjred 11012 | A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remul2d 11013 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immul2d 11014 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | redivapd 11015 | Real part of a division. Related to remul2 10914. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | imdivapd 11016 | Imaginary part of a division. Related to remul2 10914. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | crred 11017 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | crimd 11018 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cnreim 11019 | Complex apartness in terms of real and imaginary parts. See also apreim 8590 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.) |
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Theorem | caucvgrelemrec 11020* | Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.) |
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Theorem | caucvgrelemcau 11021* | Lemma for caucvgre 11022. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.) |
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Theorem | caucvgre 11022* |
Convergence of real sequences.
A Cauchy sequence (as defined here, which has a rate of convergence
built in) of real numbers converges to a real number. Specifically on
rate of convergence, all terms after the nth term must be within
(Contributed by Jim Kingdon, 19-Jul-2021.) |
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Theorem | cvg1nlemcxze 11023 | Lemma for cvg1n 11027. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.) |
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Theorem | cvg1nlemf 11024* |
Lemma for cvg1n 11027. The modified sequence ![]() |
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Theorem | cvg1nlemcau 11025* |
Lemma for cvg1n 11027. By selecting spaced out terms for the
modified
sequence ![]() ![]() ![]() ![]() ![]() |
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Theorem | cvg1nlemres 11026* |
Lemma for cvg1n 11027. The original sequence ![]() ![]() |
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Theorem | cvg1n 11027* |
Convergence of real sequences.
This is a version of caucvgre 11022 with a constant multiplier (Contributed by Jim Kingdon, 1-Aug-2021.) |
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Theorem | uzin2 11028 | The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.) |
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Theorem | rexanuz 11029* | Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.) |
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Theorem | rexfiuz 11030* | Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.) |
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Theorem | rexuz3 11031* | Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.) |
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Theorem | rexanuz2 11032* | Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.) |
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Theorem | r19.29uz 11033* | A version of 19.29 1631 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.) |
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Theorem | r19.2uz 11034* | A version of r19.2m 3524 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
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Theorem | recvguniqlem 11035 | Lemma for recvguniq 11036. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.) |
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Theorem | recvguniq 11036* | Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.) |
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Syntax | csqrt 11037 | Extend class notation to include square root of a complex number. |
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Syntax | cabs 11038 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
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Definition | df-rsqrt 11039* |
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.) |
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Definition | df-abs 11040 | Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.) |
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Theorem | sqrtrval 11041* | Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.) |
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Theorem | absval 11042 | 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.) |
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Theorem | rennim 11043 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
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Theorem | sqrt0rlem 11044 | Lemma for sqrt0 11045. (Contributed by Jim Kingdon, 26-Aug-2020.) |
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Theorem | sqrt0 11045 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | resqrexlem1arp 11046 |
Lemma for resqrex 11067. ![]() ![]() ![]() |
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Theorem | resqrexlemp1rp 11047* | Lemma for resqrex 11067. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10492 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
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Theorem | resqrexlemf 11048* | Lemma for resqrex 11067. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
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Theorem | resqrexlemf1 11049* | Lemma for resqrex 11067. 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.) |
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Theorem | resqrexlemfp1 11050* | Lemma for resqrex 11067. 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemover 11051* | Lemma for resqrex 11067. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemdec 11052* | Lemma for resqrex 11067. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemdecn 11053* | Lemma for resqrex 11067. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemlo 11054* | Lemma for resqrex 11067. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemcalc1 11055* | Lemma for resqrex 11067. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemcalc2 11056* | Lemma for resqrex 11067. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemcalc3 11057* | Lemma for resqrex 11067. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemnmsq 11058* | Lemma for resqrex 11067. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemnm 11059* | Lemma for resqrex 11067. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemcvg 11060* | Lemma for resqrex 11067. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemgt0 11061* | Lemma for resqrex 11067. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemoverl 11062* |
Lemma for resqrex 11067. Every term in the sequence is an
overestimate
compared with the limit ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemglsq 11063* |
Lemma for resqrex 11067. The sequence formed by squaring each term
of ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemga 11064* |
Lemma for resqrex 11067. The sequence formed by squaring each term
of ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemsqa 11065* |
Lemma for resqrex 11067. The square of a limit is ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrexlemex 11066* | Lemma for resqrex 11067. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | resqrex 11067* | Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rsqrmo 11068* | Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.) |
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Theorem | rersqreu 11069* | Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.) |
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Theorem | resqrtcl 11070 | Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | rersqrtthlem 11071 | Lemma for resqrtth 11072. (Contributed by Jim Kingdon, 10-Aug-2021.) |
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Theorem | resqrtth 11072 | Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | remsqsqrt 11073 | Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrtge0 11074 | The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrtgt0 11075 | The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.) |
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Theorem | sqrtmul 11076 | Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtle 11077 | Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtlt 11078 | Square root is strictly monotonic. Closed form of sqrtlti 11178. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrt11ap 11079 | Analogue to sqrt11 11080 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.) |
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Theorem | sqrt11 11080 | The square root function is one-to-one. Also see sqrt11ap 11079 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sqrt00 11081 | A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | rpsqrtcl 11082 | The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.) |
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Theorem | sqrtdiv 11083 | Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.) |
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Theorem | sqrtsq2 11084 | Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sqrtsq 11085 | Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtmsq 11086 | Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrt1 11087 | The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.) |
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Theorem | sqrt4 11088 | The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.) |
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Theorem | sqrt9 11089 | The square root of 9 is 3. (Contributed by NM, 11-May-2004.) |
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Theorem | sqrt2gt1lt2 11090 | 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | absneg 11091 | Absolute value of negative. (Contributed by NM, 27-Feb-2005.) |
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Theorem | abscl 11092 | Real closure of absolute value. (Contributed by NM, 3-Oct-1999.) |
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Theorem | abscj 11093 | 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.) |
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Theorem | absvalsq 11094 | Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.) |
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Theorem | absvalsq2 11095 | Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sqabsadd 11096 | Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) |
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Theorem | sqabssub 11097 | Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.) |
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Theorem | absval2 11098 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.) |
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Theorem | abs0 11099 | The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | absi 11100 | The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
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