Theorem List for Intuitionistic Logic Explorer - 9401-9500 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | nnsqcld 9401 |
The naturals are closed under squaring. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | nnexpcld 9402 |
Closure of exponentiation of nonnegative integers. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | nn0expcld 9403 |
Closure of exponentiation of nonnegative integers. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | rpexpcld 9404 |
Closure law for exponentiation of positive reals. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | reexpclzapd 9405 |
Closure of exponentiation of reals. (Contributed by Jim Kingdon,
13-Jun-2020.)
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Theorem | resqcld 9406 |
Closure of square in reals. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | sqge0d 9407 |
A square of a real is nonnegative. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | sqgt0apd 9408 |
The square of a real apart from zero is positive. (Contributed by Jim
Kingdon, 13-Jun-2020.)
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Theorem | leexp2ad 9409 |
Ordering relationship for exponentiation. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | leexp2rd 9410 |
Ordering relationship for exponentiation. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | lt2sqd 9411 |
The square function on nonnegative reals is strictly monotonic.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | le2sqd 9412 |
The square function on nonnegative reals is monotonic. (Contributed by
Mario Carneiro, 28-May-2016.)
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Theorem | sq11d 9413 |
The square function is one-to-one for nonnegative reals. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | sq11ap 9414 |
Analogue to sq11 9326 but for apartness. (Contributed by Jim
Kingdon,
12-Aug-2021.)
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       #     #
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3.7 Elementary real and complex
functions
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3.7.1 The "shift" operation
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Syntax | cshi 9415 |
Extend class notation with function shifter.
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Definition | df-shft 9416* |
Define a function shifter. This operation offsets the value argument of
a function (ordinarily on a subset of ) and produces a new
function on .
See shftval 9426 for its value. (Contributed by NM,
20-Jul-2005.)
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Theorem | shftlem 9417* |
Two ways to write a shifted set   . (Contributed by Mario
Carneiro, 3-Nov-2013.)
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Theorem | shftuz 9418* |
A shift of the upper integers. (Contributed by Mario Carneiro,
5-Nov-2013.)
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Theorem | shftfvalg 9419* |
The value of the sequence shifter operation is a function on .
is ordinarily
an integer. (Contributed by NM, 20-Jul-2005.)
(Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | ovshftex 9420 |
Existence of the result of applying shift. (Contributed by Jim Kingdon,
15-Aug-2021.)
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Theorem | shftfibg 9421 |
Value of a fiber of the relation . (Contributed by Jim Kingdon,
15-Aug-2021.)
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Theorem | shftfval 9422* |
The value of the sequence shifter operation is a function on .
is ordinarily
an integer. (Contributed by NM, 20-Jul-2005.)
(Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | shftdm 9423* |
Domain of a relation shifted by . The set on the right is more
commonly notated as  
(meaning add to every
element of ).
(Contributed by Mario Carneiro, 3-Nov-2013.)
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Theorem | shftfib 9424 |
Value of a fiber of the relation . (Contributed by Mario
Carneiro, 4-Nov-2013.)
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Theorem | shftfn 9425* |
Functionality and domain of a sequence shifted by . (Contributed
by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
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Theorem | shftval 9426 |
Value of a sequence shifted by . (Contributed by NM,
20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
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Theorem | shftval2 9427 |
Value of a sequence shifted by . (Contributed by NM,
20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftval3 9428 |
Value of a sequence shifted by . (Contributed by NM,
20-Jul-2005.)
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Theorem | shftval4 9429 |
Value of a sequence shifted by  .
(Contributed by NM,
18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftval5 9430 |
Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftf 9431* |
Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | 2shfti 9432 |
Composite shift operations. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftidt2 9433 |
Identity law for the shift operation. (Contributed by Mario Carneiro,
5-Nov-2013.)
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Theorem | shftidt 9434 |
Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.)
(Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftcan1 9435 |
Cancellation law for the shift operation. (Contributed by NM,
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftcan2 9436 |
Cancellation law for the shift operation. (Contributed by NM,
4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
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Theorem | shftvalg 9437 |
Value of a sequence shifted by . (Contributed by Scott Fenton,
16-Dec-2017.)
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Theorem | shftval4g 9438 |
Value of a sequence shifted by  .
(Contributed by Jim Kingdon,
19-Aug-2021.)
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3.7.2 Real and imaginary parts;
conjugate
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Syntax | ccj 9439 |
Extend class notation to include complex conjugate function.
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Syntax | cre 9440 |
Extend class notation to include real part of a complex number.
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Syntax | cim 9441 |
Extend class notation to include imaginary part of a complex number.
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Definition | df-cj 9442* |
Define the complex conjugate function. See cjcli 9513 for its closure and
cjval 9445 for its value. (Contributed by NM,
9-May-1999.) (Revised by
Mario Carneiro, 6-Nov-2013.)
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Definition | df-re 9443 |
Define a function whose value is the real part of a complex number. See
reval 9449 for its value, recli 9511 for its closure, and replim 9459 for its use
in decomposing a complex number. (Contributed by NM, 9-May-1999.)
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Definition | df-im 9444 |
Define a function whose value is the imaginary part of a complex number.
See imval 9450 for its value, imcli 9512 for its closure, and replim 9459 for its
use in decomposing a complex number. (Contributed by NM,
9-May-1999.)
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Theorem | cjval 9445* |
The value of the conjugate of a complex number. (Contributed by Mario
Carneiro, 6-Nov-2013.)
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Theorem | cjth 9446 |
The defining property of the complex conjugate. (Contributed by Mario
Carneiro, 6-Nov-2013.)
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Theorem | cjf 9447 |
Domain and codomain of the conjugate function. (Contributed by Mario
Carneiro, 6-Nov-2013.)
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Theorem | cjcl 9448 |
The conjugate of a complex number is a complex number (closure law).
(Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
6-Nov-2013.)
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Theorem | reval 9449 |
The value of the real part of a complex number. (Contributed by NM,
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | imval 9450 |
The value of the imaginary part of a complex number. (Contributed by
NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | imre 9451 |
The imaginary part of a complex number in terms of the real part
function. (Contributed by NM, 12-May-2005.) (Revised by Mario
Carneiro, 6-Nov-2013.)
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Theorem | reim 9452 |
The real part of a complex number in terms of the imaginary part
function. (Contributed by Mario Carneiro, 31-Mar-2015.)
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Theorem | recl 9453 |
The real part of a complex number is real. (Contributed by NM,
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | imcl 9454 |
The imaginary part of a complex number is real. (Contributed by NM,
9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | ref 9455 |
Domain and codomain of the real part function. (Contributed by Paul
Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | imf 9456 |
Domain and codomain of the imaginary part function. (Contributed by
Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
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Theorem | crre 9457 |
The real part of a complex number representation. Definition 10-3.1 of
[Gleason] p. 132. (Contributed by NM,
12-May-2005.) (Revised by Mario
Carneiro, 7-Nov-2013.)
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Theorem | crim 9458 |
The real part of a complex number representation. Definition 10-3.1 of
[Gleason] p. 132. (Contributed by NM,
12-May-2005.) (Revised by Mario
Carneiro, 7-Nov-2013.)
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Theorem | replim 9459 |
Reconstruct a complex number from its real and imaginary parts.
(Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
7-Nov-2013.)
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Theorem | remim 9460 |
Value of the conjugate of a complex number. The value is the real part
minus times
the imaginary part. Definition 10-3.2 of [Gleason]
p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro,
7-Nov-2013.)
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Theorem | reim0 9461 |
The imaginary part of a real number is 0. (Contributed by NM,
18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
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Theorem | reim0b 9462 |
A number is real iff its imaginary part is 0. (Contributed by NM,
26-Sep-2005.)
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Theorem | rereb 9463 |
A number is real iff it equals its real part. Proposition 10-3.4(f) of
[Gleason] p. 133. (Contributed by NM,
20-Aug-2008.)
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Theorem | mulreap 9464 |
A product with a real multiplier apart from zero is real iff the
multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
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  #  
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Theorem | rere 9465 |
A real number equals its real part. One direction of Proposition
10-3.4(f) of [Gleason] p. 133.
(Contributed by Paul Chapman,
7-Sep-2007.)
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Theorem | cjreb 9466 |
A number is real iff it equals its complex conjugate. Proposition
10-3.4(f) of [Gleason] p. 133.
(Contributed by NM, 2-Jul-2005.) (Revised
by Mario Carneiro, 14-Jul-2014.)
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Theorem | recj 9467 |
Real part of a complex conjugate. (Contributed by Mario Carneiro,
14-Jul-2014.)
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Theorem | reneg 9468 |
Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by
Mario Carneiro, 14-Jul-2014.)
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Theorem | readd 9469 |
Real part distributes over addition. (Contributed by NM, 17-Mar-2005.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | resub 9470 |
Real part distributes over subtraction. (Contributed by NM,
17-Mar-2005.)
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Theorem | remullem 9471 |
Lemma for remul 9472, immul 9479, and cjmul 9485. (Contributed by NM,
28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | remul 9472 |
Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by
Mario Carneiro, 14-Jul-2014.)
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Theorem | remul2 9473 |
Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
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Theorem | redivap 9474 |
Real part of a division. Related to remul2 9473. (Contributed by Jim
Kingdon, 14-Jun-2020.)
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  #                |
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Theorem | imcj 9475 |
Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | imneg 9476 |
The imaginary part of a negative number. (Contributed by NM,
18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | imadd 9477 |
Imaginary part distributes over addition. (Contributed by NM,
18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | imsub 9478 |
Imaginary part distributes over subtraction. (Contributed by NM,
18-Mar-2005.)
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Theorem | immul 9479 |
Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised
by Mario Carneiro, 14-Jul-2014.)
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Theorem | immul2 9480 |
Imaginary part of a product. (Contributed by Mario Carneiro,
2-Aug-2014.)
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Theorem | imdivap 9481 |
Imaginary part of a division. Related to immul2 9480. (Contributed by Jim
Kingdon, 14-Jun-2020.)
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  #                |
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Theorem | cjre 9482 |
A real number equals its complex conjugate. Proposition 10-3.4(f) of
[Gleason] p. 133. (Contributed by NM,
8-Oct-1999.)
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Theorem | cjcj 9483 |
The conjugate of the conjugate is the original complex number.
Proposition 10-3.4(e) of [Gleason] p. 133.
(Contributed by NM,
29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjadd 9484 |
Complex conjugate distributes over addition. Proposition 10-3.4(a) of
[Gleason] p. 133. (Contributed by NM,
31-Jul-1999.) (Revised by Mario
Carneiro, 14-Jul-2014.)
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Theorem | cjmul 9485 |
Complex conjugate distributes over multiplication. Proposition 10-3.4(c)
of [Gleason] p. 133. (Contributed by NM,
29-Jul-1999.) (Proof shortened
by Mario Carneiro, 14-Jul-2014.)
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Theorem | ipcnval 9486 |
Standard inner product on complex numbers. (Contributed by NM,
29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjmulrcl 9487 |
A complex number times its conjugate is real. (Contributed by NM,
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjmulval 9488 |
A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjmulge0 9489 |
A complex number times its conjugate is nonnegative. (Contributed by NM,
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjneg 9490 |
Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | addcj 9491 |
A number plus its conjugate is twice its real part. Compare Proposition
10-3.4(h) of [Gleason] p. 133.
(Contributed by NM, 21-Jan-2007.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | cjsub 9492 |
Complex conjugate distributes over subtraction. (Contributed by NM,
28-Apr-2005.)
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Theorem | cjexp 9493 |
Complex conjugate of positive integer exponentiation. (Contributed by
NM, 7-Jun-2006.)
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Theorem | imval2 9494 |
The imaginary part of a number in terms of complex conjugate.
(Contributed by NM, 30-Apr-2005.)
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Theorem | re0 9495 |
The real part of zero. (Contributed by NM, 27-Jul-1999.)
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Theorem | im0 9496 |
The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
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Theorem | re1 9497 |
The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
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Theorem | im1 9498 |
The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
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Theorem | rei 9499 |
The real part of .
(Contributed by Scott Fenton, 9-Jun-2006.)
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Theorem | imi 9500 |
The imaginary part of . (Contributed by Scott Fenton,
9-Jun-2006.)
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