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Type | Label | Description |
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Statement | ||
Theorem | cjap 10401 | Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.) |
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Theorem | cjap0 10402 | A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.) |
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Theorem | cjne0 10403 | A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10402 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.) |
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Theorem | cjdivap 10404 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.) |
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Theorem | cnrecnv 10405* |
The inverse to the canonical bijection from ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | recli 10406 | The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
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Theorem | imcli 10407 | The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
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Theorem | cjcli 10408 | Closure law for complex conjugate. (Contributed by NM, 11-May-1999.) |
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Theorem | replimi 10409 | Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.) |
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Theorem | cjcji 10410 | The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.) |
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Theorem | reim0bi 10411 | A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.) |
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Theorem | rerebi 10412 | A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.) |
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Theorem | cjrebi 10413 | A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) |
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Theorem | recji 10414 | Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | imcji 10415 | Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjmulrcli 10416 | A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.) |
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Theorem | cjmulvali 10417 | A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjmulge0i 10418 | A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.) |
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Theorem | renegi 10419 | Real part of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | imnegi 10420 | Imaginary part of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | cjnegi 10421 | Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | addcji 10422 | A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.) |
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Theorem | readdi 10423 | Real part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
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Theorem | imaddi 10424 | Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
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Theorem | remuli 10425 | Real part of a product. (Contributed by NM, 28-Jul-1999.) |
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Theorem | immuli 10426 | Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) |
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Theorem | cjaddi 10427 | Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) |
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Theorem | cjmuli 10428 | Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) |
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Theorem | ipcni 10429 | Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjdivapi 10430 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.) |
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Theorem | crrei 10431 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
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Theorem | crimi 10432 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
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Theorem | recld 10433 | The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imcld 10434 | The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjcld 10435 | Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | replimd 10436 | Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remimd 10437 |
Value of the conjugate of a complex number. The value is the real part
minus ![]() |
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Theorem | cjcjd 10438 | The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | reim0bd 10439 | A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | rerebd 10440 | A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjrebd 10441 | A number is real iff it 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 | cjne0d 10442 | A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10443 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjap0d 10443 | A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.) |
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Theorem | recjd 10444 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imcjd 10445 | Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulrcld 10446 | A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulvald 10447 | A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulge0d 10448 | A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | renegd 10449 | Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imnegd 10450 | Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjnegd 10451 | Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | addcjd 10452 | A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjexpd 10453 | Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | readdd 10454 | Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imaddd 10455 | Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | resubd 10456 | Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imsubd 10457 | Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remuld 10458 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immuld 10459 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjaddd 10460 | 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 10461 | 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 10462 | Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjdivapd 10463 | Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | rered 10464 | 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 10465 | The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjred 10466 | 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 10467 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immul2d 10468 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | redivapd 10469 | Real part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | imdivapd 10470 | Imaginary part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | crred 10471 | 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 10472 | 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 | caucvgrelemrec 10473* | Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.) |
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Theorem | caucvgrelemcau 10474* | Lemma for caucvgre 10475. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.) |
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Theorem | caucvgre 10475* |
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 10476 | Lemma for cvg1n 10480. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.) |
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Theorem | cvg1nlemf 10477* |
Lemma for cvg1n 10480. The modified sequence ![]() |
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Theorem | cvg1nlemcau 10478* |
Lemma for cvg1n 10480. By selecting spaced out terms for the
modified
sequence ![]() ![]() ![]() ![]() ![]() |
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Theorem | cvg1nlemres 10479* |
Lemma for cvg1n 10480. The original sequence ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cvg1n 10480* |
Convergence of real sequences.
This is a version of caucvgre 10475 with a constant multiplier (Contributed by Jim Kingdon, 1-Aug-2021.) |
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Theorem | uzin2 10481 | The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.) |
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Theorem | rexanuz 10482* | Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.) |
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Theorem | rexfiuz 10483* | Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.) |
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Theorem | rexuz3 10484* | 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 10485* | Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.) |
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Theorem | r19.29uz 10486* | A version of 19.29 1557 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.) |
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Theorem | r19.2uz 10487* | A version of r19.2m 3373 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
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Theorem | recvguniqlem 10488 | Lemma for recvguniq 10489. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.) |
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Theorem | recvguniq 10489* | Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.) |
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Syntax | csqrt 10490 | Extend class notation to include square root of a complex number. |
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Syntax | cabs 10491 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
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Definition | df-rsqrt 10492* |
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 10493 | Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.) |
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Theorem | sqrtrval 10494* | Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.) |
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Theorem | absval 10495 | 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 10496 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
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Theorem | sqrt0rlem 10497 | Lemma for sqrt0 10498. (Contributed by Jim Kingdon, 26-Aug-2020.) |
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Theorem | sqrt0 10498 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | resqrexlem1arp 10499 |
Lemma for resqrex 10520. ![]() ![]() ![]() |
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Theorem | resqrexlemp1rp 10500* | Lemma for resqrex 10520. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 9941 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
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