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Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremim1 10701 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 10702 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 10703 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremcj0 10704 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 10705 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 10706 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 10707 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 10708 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

Theoremcjap 10709 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjap0 10710 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjne0 10711 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10710 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)

Theoremcjdivap 10712 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcnrecnv 10713* The inverse to the canonical bijection from to from cnref1o 9468. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremrecli 10714 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 10715 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 10716 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 10717 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 10718 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 10719 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

Theoremrerebi 10720 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)

Theoremcjrebi 10721 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.)

Theoremrecji 10722 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremimcji 10723 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulrcli 10724 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)

Theoremcjmulvali 10725 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulge0i 10726 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)

Theoremrenegi 10727 Real part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremimnegi 10728 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremcjnegi 10729 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)

Theoremaddcji 10730 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.)

Theoremreaddi 10731 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremimaddi 10732 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremremuli 10733 Real part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremimmuli 10734 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremcjaddi 10735 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremcjmuli 10736 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremipcni 10737 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)

Theoremcjdivapi 10738 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcrrei 10739 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremcrimi 10740 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremrecld 10741 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcld 10742 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjcld 10743 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreplimd 10744 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremimd 10745 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 Mario Carneiro, 29-May-2016.)

Theoremcjcjd 10746 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.)

Theoremreim0bd 10747 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrerebd 10748 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjrebd 10749 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.)

Theoremcjne0d 10750 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10751 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjap0d 10751 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
#        #

Theoremrecjd 10752 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcjd 10753 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulrcld 10754 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulvald 10755 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulge0d 10756 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrenegd 10757 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimnegd 10758 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjnegd 10759 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremaddcjd 10760 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.)

Theoremcjexpd 10761 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreaddd 10762 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimaddd 10763 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresubd 10764 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimsubd 10765 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremuld 10766 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmuld 10767 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjaddd 10768 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmuld 10769 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremipcnd 10770 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjdivapd 10771 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremrered 10772 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.)

Theoremreim0d 10773 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjred 10774 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremul2d 10775 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmul2d 10776 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremredivapd 10777 Real part of a division. Related to remul2 10676. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremimdivapd 10778 Imaginary part of a division. Related to remul2 10676. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremcrred 10779 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrimd 10780 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcnreim 10781 Complex apartness in terms of real and imaginary parts. See also apreim 8388 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
# # #

4.7.3  Sequence convergence

Theoremcaucvgrelemrec 10782* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
#

Theoremcaucvgrelemcau 10783* Lemma for caucvgre 10784. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)

Theoremcaucvgre 10784* 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 of the nth term.

(Contributed by Jim Kingdon, 19-Jul-2021.)

Theoremcvg1nlemcxze 10785 Lemma for cvg1n 10789. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)

Theoremcvg1nlemf 10786* Lemma for cvg1n 10789. The modified sequence is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemcau 10787* Lemma for cvg1n 10789. By selecting spaced out terms for the modified sequence , the terms are within (without the constant ). (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemres 10788* Lemma for cvg1n 10789. The original sequence has a limit (turns out it is the same as the limit of the modified sequence ). (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1n 10789* Convergence of real sequences.

This is a version of caucvgre 10784 with a constant multiplier on the rate of convergence. That is, all terms after the nth term must be within of the nth term.

(Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremuzin2 10790 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 10791* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexfiuz 10792* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremrexuz3 10793* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 10794* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 10795* A version of 19.29 1600 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 10796* A version of r19.2m 3453 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrecvguniqlem 10797 Lemma for recvguniq 10798. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)

Theoremrecvguniq 10798* Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)

4.7.4  Square root; absolute value

Syntaxcsqrt 10799 Extend class notation to include square root of a complex number.

Syntaxcabs 10800 Extend class notation to include a function for the absolute value (modulus) of a complex number.

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