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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcjap 10401 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
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Theoremcjap0 10402 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
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Theoremcjne0 10403 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10402 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)

Theoremcjdivap 10404 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
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Theoremcnrecnv 10405* The inverse to the canonical bijection from to from cnref1o 9194. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremrecli 10406 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 10407 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 10408 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 10409 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 10410 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 10411 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

Theoremrerebi 10412 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)

Theoremcjrebi 10413 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 10414 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremimcji 10415 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulrcli 10416 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)

Theoremcjmulvali 10417 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulge0i 10418 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)

Theoremrenegi 10419 Real part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremimnegi 10420 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremcjnegi 10421 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)

Theoremaddcji 10422 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 10425 Real part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremimmuli 10426 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremcjaddi 10427 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremcjmuli 10428 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremipcni 10429 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)

Theoremcjdivapi 10430 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcrrei 10431 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremcrimi 10432 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremrecld 10433 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcld 10434 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjcld 10435 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreplimd 10436 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremimd 10437 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 10438 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 10439 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrerebd 10440 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjrebd 10441 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 10442 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10443 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjap0d 10443 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
#        #

Theoremrecjd 10444 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcjd 10445 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulrcld 10446 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulvald 10447 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulge0d 10448 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrenegd 10449 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimnegd 10450 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjnegd 10451 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremaddcjd 10452 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 10453 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreaddd 10454 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimaddd 10455 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresubd 10456 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimsubd 10457 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremuld 10458 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmuld 10459 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjaddd 10460 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmuld 10461 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremipcnd 10462 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjdivapd 10463 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
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Theoremrered 10464 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 10465 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjred 10466 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremul2d 10467 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmul2d 10468 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremredivapd 10469 Real part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 15-Jun-2020.)
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Theoremimdivapd 10470 Imaginary part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 15-Jun-2020.)
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Theoremcrred 10471 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrimd 10472 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 10473* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
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Theoremcaucvgrelemcau 10474* Lemma for caucvgre 10475. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)

Theoremcaucvgre 10475* 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 10476 Lemma for cvg1n 10480. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)

Theoremcvg1nlemf 10477* Lemma for cvg1n 10480. The modified sequence is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemcau 10478* Lemma for cvg1n 10480. By selecting spaced out terms for the modified sequence , the terms are within (without the constant ). (Contributed by Jim Kingdon, 1-Aug-2021.)

Theoremcvg1nlemres 10479* Lemma for cvg1n 10480. 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 10480* Convergence of real sequences.

This is a version of caucvgre 10475 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 10481 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 10482* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexfiuz 10483* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremrexuz3 10484* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 10485* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 10486* A version of 19.29 1557 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 10487* A version of r19.2m 3373 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrecvguniqlem 10488 Lemma for recvguniq 10489. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)

Theoremrecvguniq 10489* Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)

3.7.4  Square root; absolute value

Syntaxcsqrt 10490 Extend class notation to include square root of a complex number.

Syntaxcabs 10491 Extend class notation to include a function for the absolute value (modulus) of a complex number.

Definitiondf-rsqrt 10492* 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 10493 Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtrval 10494* Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)

Theoremabsval 10495 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 10496 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremsqrt0rlem 10497 Lemma for sqrt0 10498. (Contributed by Jim Kingdon, 26-Aug-2020.)

Theoremsqrt0 10498 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrexlem1arp 10499 Lemma for resqrex 10520. is a positive real (expressed in a way that will help apply seqf 9941 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)

Theoremresqrexlemp1rp 10500* Lemma for resqrex 10520. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 9941 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)

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