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Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshftuz 10601* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfvalg 10602* 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 10603 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfibg 10604 Value of a fiber of the relation . (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfval 10605* 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 10606* 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 10607 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 10608* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 10609 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 10610 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 10611 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 10612 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 10613 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 10614* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 10615 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 10616 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 10617 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 10618 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 10619 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftvalg 10620 Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremshftval4g 10621 Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)

Theoremseq3shft 10622* Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)

4.7.2  Real and imaginary parts; conjugate

Syntaxccj 10623 Extend class notation to include complex conjugate function.

Syntaxcre 10624 Extend class notation to include real part of a complex number.

Syntaxcim 10625 Extend class notation to include imaginary part of a complex number.

Definitiondf-cj 10626* Define the complex conjugate function. See cjcli 10697 for its closure and cjval 10629 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Definitiondf-re 10627 Define a function whose value is the real part of a complex number. See reval 10633 for its value, recli 10695 for its closure, and replim 10643 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Definitiondf-im 10628 Define a function whose value is the imaginary part of a complex number. See imval 10634 for its value, imcli 10696 for its closure, and replim 10643 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Theoremcjval 10629* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 10630 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 10631 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 10632 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 10633 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 10634 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 10635 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 10636 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 10637 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 10638 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 10639 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 10640 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 10641 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 10642 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 10643 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremremim 10644 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 10645 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0b 10646 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)

Theoremrereb 10647 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 10648 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremrere 10649 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 10650 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 10651 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremreneg 10652 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremreadd 10653 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremresub 10654 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Theoremremullem 10655 Lemma for remul 10656, immul 10663, and cjmul 10669. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul 10656 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul2 10657 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremredivap 10658 Real part of a division. Related to remul2 10657. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremimcj 10659 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimneg 10660 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimadd 10661 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimsub 10662 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)

Theoremimmul 10663 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimmul2 10664 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremimdivap 10665 Imaginary part of a division. Related to immul2 10664. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcjre 10666 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)

Theoremcjcj 10667 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 10668 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 10669 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 10670 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulrcl 10671 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulval 10672 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulge0 10673 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjneg 10674 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremaddcj 10675 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 10676 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)

Theoremcjexp 10677 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)

Theoremimval2 10678 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)

Theoremre0 10679 The real part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremim0 10680 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremre1 10681 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremim1 10682 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 10683 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 10684 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremcj0 10685 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 10686 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 10687 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 10688 The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremcj11 10689 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

Theoremcjap 10690 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjap0 10691 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjne0 10692 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10691 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)

Theoremcjdivap 10693 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcnrecnv 10694* The inverse to the canonical bijection from to from cnref1o 9452. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremrecli 10695 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 10696 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 10697 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 10698 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 10699 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.)

Theoremreim0bi 10700 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

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