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Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcj0 9501 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 9502 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 9503 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 9504 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 9505 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

Theoremcjap 9506 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjap0 9507 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theoremcjne0 9508 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 9507 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)

Theoremcjdivap 9509 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcnrecnv 9510* The inverse to the canonical bijection from to from cnref1o 8582. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremrecli 9511 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 9512 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 9513 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 9514 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 9515 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 9516 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

Theoremrerebi 9517 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)

Theoremcjrebi 9518 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 9519 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremimcji 9520 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulrcli 9521 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)

Theoremcjmulvali 9522 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulge0i 9523 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)

Theoremrenegi 9524 Real part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremimnegi 9525 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremcjnegi 9526 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)

Theoremaddcji 9527 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.)

Theoremremuli 9530 Real part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremimmuli 9531 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremcjaddi 9532 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremcjmuli 9533 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremipcni 9534 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)

Theoremcjdivapi 9535 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcrrei 9536 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremcrimi 9537 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremrecld 9538 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcld 9539 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjcld 9540 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreplimd 9541 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremimd 9542 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 9543 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 9544 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrerebd 9545 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjrebd 9546 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 9547 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 9548 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjap0d 9548 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
#        #

Theoremrecjd 9549 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcjd 9550 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulrcld 9551 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulvald 9552 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulge0d 9553 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrenegd 9554 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimnegd 9555 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjnegd 9556 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremaddcjd 9557 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 9558 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreaddd 9559 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimaddd 9560 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresubd 9561 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimsubd 9562 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremuld 9563 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmuld 9564 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjaddd 9565 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmuld 9566 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremipcnd 9567 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjdivapd 9568 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremrered 9569 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 9570 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjred 9571 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremul2d 9572 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmul2d 9573 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremredivapd 9574 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremimdivapd 9575 Imaginary part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
#

Theoremcrred 9576 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrimd 9577 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

3.7.3  Sequence convergence

Theoremcaucvgrelemrec 9578* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
#

Theoremcaucvgrelemcau 9579* Lemma for caucvgre 9580. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)

Theoremcaucvgre 9580* 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 9581 Lemma for cvg1n 9585. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)

Theoremcvg1nlemf 9582* Lemma for cvg1n 9585. The modified sequence is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemcau 9583* Lemma for cvg1n 9585. By selecting spaced out terms for the modified sequence , the terms are within (without the constant ). (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemres 9584* Lemma for cvg1n 9585. 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 9585* Convergence of real sequences.

This is a version of caucvgre 9580 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 9586 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 9587* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexuz3 9588* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 9589* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 9590* A version of 19.29 1511 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 9591* A version of r19.2m 3309 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrecvguniqlem 9592 Lemma for recvguniq 9593. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)

Theoremrecvguniq 9593* Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)

3.7.4  Square root; absolute value

Syntaxcsqrt 9594 Extend class notation to include square root of a complex number.

Syntaxcabs 9595 Extend class notation to include a function for the absolute value (modulus) of a complex number.

Definitiondf-rsqrt 9596* 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.)

Definitiondf-abs 9597 Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtrval 9598* Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)

Theoremabsval 9599 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.)

Theoremrennim 9600 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

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