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Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnsqcld 9401 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnexpcld 9402 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnn0expcld 9403 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpexpcld 9404 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreexpclzapd 9405 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremresqcld 9406 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqge0d 9407 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsqgt0apd 9408 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
#

Theoremleexp2ad 9409 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremleexp2rd 9410 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2sqd 9411 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremle2sqd 9412 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11d 9413 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsq11ap 9414 Analogue to sq11 9326 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
# #

3.7  Elementary real and complex functions

3.7.1  The "shift" operation

Syntaxcshi 9415 Extend class notation with function shifter.

Definitiondf-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.)

Theoremshftlem 9417* Two ways to write a shifted set . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftuz 9418* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfvalg 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.)

Theoremovshftex 9420 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfibg 9421 Value of a fiber of the relation . (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfval 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.)

Theoremshftdm 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.)

Theoremshftfib 9424 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 9425* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 9426 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 9427 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 9428 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 9429 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 9430 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 9431* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 9432 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 9433 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 9434 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 9435 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 9436 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftvalg 9437 Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremshftval4g 9438 Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)

3.7.2  Real and imaginary parts; conjugate

Syntaxccj 9439 Extend class notation to include complex conjugate function.

Syntaxcre 9440 Extend class notation to include real part of a complex number.

Syntaxcim 9441 Extend class notation to include imaginary part of a complex number.

Definitiondf-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.)

Definitiondf-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.)

Definitiondf-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.)

Theoremcjval 9445* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 9446 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 9447 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 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.)

Theoremreval 9449 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 9450 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 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.)

Theoremreim 9452 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 9453 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 9454 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 9455 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 9456 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 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.)

Theoremcrim 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.)

Theoremreplim 9459 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremremim 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.)

Theoremreim0 9461 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0b 9462 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)

Theoremrereb 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.)

Theoremmulreap 9464 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremrere 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.)

Theoremcjreb 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.)

Theoremrecj 9467 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremreneg 9468 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremreadd 9469 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremresub 9470 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Theoremremullem 9471 Lemma for remul 9472, immul 9479, and cjmul 9485. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul 9472 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul2 9473 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremredivap 9474 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremimcj 9475 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimneg 9476 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimadd 9477 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimsub 9478 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)

Theoremimmul 9479 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimmul2 9480 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremimdivap 9481 Imaginary part of a division. Related to immul2 9480. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcjre 9482 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)

Theoremcjcj 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.)

Theoremcjadd 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.)

Theoremcjmul 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.)

Theoremipcnval 9486 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulrcl 9487 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulval 9488 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulge0 9489 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjneg 9490 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremaddcj 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.)

Theoremcjsub 9492 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)

Theoremcjexp 9493 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)

Theoremimval2 9494 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)

Theoremre0 9495 The real part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremim0 9496 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremre1 9497 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremim1 9498 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 9499 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 9500 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

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