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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnfzo0hash 10301 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
..^

Theoremhashfacen 10302* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremleisorel 10303 Version of isorel 5601 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremzfz1isolemsplit 10304 Lemma for zfz1iso 10307. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)

Theoremzfz1isolemiso 10305* Lemma for zfz1iso 10307. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

Theoremzfz1isolem1 10306* Lemma for zfz1iso 10307. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

Theoremzfz1iso 10307* A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

Theoremiseqcoll 10308* The function contains a sparse set of nonzero values to be summed. The function is an order isomorphism from the set of nonzero values of to a 1-based finite sequence, and collects these nonzero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 2-Apr-2014.)

3.7  Elementary real and complex functions

3.7.1  The "shift" operation

Syntaxcshi 10309 Extend class notation with function shifter.

Definitiondf-shft 10310* 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 10320 for its value. (Contributed by NM, 20-Jul-2005.)

Theoremshftlem 10311* Two ways to write a shifted set . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftuz 10312* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfvalg 10313* 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 10314 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfibg 10315 Value of a fiber of the relation . (Contributed by Jim Kingdon, 15-Aug-2021.)

Theoremshftfval 10316* 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 10317* 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 10318 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 10319* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 10320 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 10321 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 10322 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 10323 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 10324 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 10325* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 10326 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 10327 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 10328 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 10329 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 10330 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftvalg 10331 Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremshftval4g 10332 Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)

Theoremseq3shft 10333* Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)

3.7.2  Real and imaginary parts; conjugate

Syntaxccj 10334 Extend class notation to include complex conjugate function.

Syntaxcre 10335 Extend class notation to include real part of a complex number.

Syntaxcim 10336 Extend class notation to include imaginary part of a complex number.

Definitiondf-cj 10337* Define the complex conjugate function. See cjcli 10408 for its closure and cjval 10340 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Definitiondf-re 10338 Define a function whose value is the real part of a complex number. See reval 10344 for its value, recli 10406 for its closure, and replim 10354 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Definitiondf-im 10339 Define a function whose value is the imaginary part of a complex number. See imval 10345 for its value, imcli 10407 for its closure, and replim 10354 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Theoremcjval 10340* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 10341 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 10342 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 10343 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 10344 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 10345 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 10346 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 10347 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 10348 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 10349 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 10350 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 10351 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 10352 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 10353 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 10354 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremremim 10355 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 10356 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0b 10357 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)

Theoremrereb 10358 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 10359 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremrere 10360 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 10361 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 10362 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremreneg 10363 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremreadd 10364 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremresub 10365 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Theoremremullem 10366 Lemma for remul 10367, immul 10374, and cjmul 10380. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul 10367 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul2 10368 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremredivap 10369 Real part of a division. Related to remul2 10368. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremimcj 10370 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimneg 10371 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimadd 10372 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimsub 10373 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)

Theoremimmul 10374 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimmul2 10375 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremimdivap 10376 Imaginary part of a division. Related to immul2 10375. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theoremcjre 10377 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)

Theoremcjcj 10378 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 10379 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 10380 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 10381 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulrcl 10382 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulval 10383 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulge0 10384 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjneg 10385 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremaddcj 10386 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 10387 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)

Theoremcjexp 10388 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)

Theoremimval2 10389 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)

Theoremre0 10390 The real part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremim0 10391 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremre1 10392 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremim1 10393 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 10394 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 10395 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremcj0 10396 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 10397 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 10398 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 10399 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 10400 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

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