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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcji 10301 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( * `  _i )  =  -u _i
 
Theoremcjreim 10302 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B ) ) )
 
Theoremcjreim2 10303 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  -  ( _i  x.  B ) ) )  =  ( A  +  ( _i  x.  B ) ) )
 
Theoremcj11 10304 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `
  A )  =  ( * `  B ) 
 <->  A  =  B ) )
 
Theoremcjap 10305 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `
  A ) #  ( * `  B )  <->  A #  B ) )
 
Theoremcjap0 10306 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( A  e.  CC  ->  ( A #  0  <->  ( * `  A ) #  0 )
 )
 
Theoremcjne0 10307 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10306 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
 |-  ( A  e.  CC  ->  ( A  =/=  0  <->  ( * `  A )  =/=  0 ) )
 
Theoremcjdivap 10308 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( * `  ( A 
 /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcnrecnv 10309* The inverse to the canonical bijection from  ( RR  X.  RR ) to  CC from cnref1o 9102. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  `' F  =  ( z  e.  CC  |->  <.
 ( Re `  z
 ) ,  ( Im
 `  z ) >. )
 
Theoremrecli 10310 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  A )  e.  RR
 
Theoremimcli 10311 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  A )  e.  RR
 
Theoremcjcli 10312 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  A )  e.  CC
 
Theoremreplimi 10313 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   =>    |-  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) )
 
Theoremcjcji 10314 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.)
 |-  A  e.  CC   =>    |-  ( * `  ( * `  A ) )  =  A
 
Theoremreim0bi 10315 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( Im `  A )  =  0 )
 
Theoremrerebi 10316 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( Re `  A )  =  A )
 
Theoremcjrebi 10317 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.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( * `  A )  =  A )
 
Theoremrecji 10318 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  ( * `  A ) )  =  ( Re
 `  A )
 
Theoremimcji 10319 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  ( * `  A ) )  =  -u ( Im `  A )
 
Theoremcjmulrcli 10320 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  e.  RR
 
Theoremcjmulvali 10321 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  =  ( ( ( Re `  A ) ^ 2 )  +  ( ( Im `  A ) ^ 2
 ) )
 
Theoremcjmulge0i 10322 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( A  x.  ( * `  A ) )
 
Theoremrenegi 10323 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  -u A )  =  -u ( Re `  A )
 
Theoremimnegi 10324 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  -u A )  =  -u ( Im `  A )
 
Theoremcjnegi 10325 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( * `  -u A )  =  -u ( * `  A )
 
Theoremaddcji 10326 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.)
 |-  A  e.  CC   =>    |-  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) )
 
Theoremreaddi 10327 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  +  B )
 )  =  ( ( Re `  A )  +  ( Re `  B ) )
 
Theoremimaddi 10328 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Im `  ( A  +  B )
 )  =  ( ( Im `  A )  +  ( Im `  B ) )
 
Theoremremuli 10329 Real part of a product. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im `  A )  x.  ( Im `  B ) ) )
 
Theoremimmuli 10330 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Im `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im `  A )  x.  ( Re `  B ) ) )
 
Theoremcjaddi 10331 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( * `  ( A  +  B )
 )  =  ( ( * `  A )  +  ( * `  B ) )
 
Theoremcjmuli 10332 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) )
 
Theoremipcni 10333 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) )
 
Theoremcjdivapi 10334 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( * `  ( A 
 /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcrrei 10335 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A
 
Theoremcrimi 10336 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B
 
Theoremrecld 10337 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  e. 
 RR )
 
Theoremimcld 10338 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  A )  e. 
 RR )
 
Theoremcjcld 10339 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  e. 
 CC )
 
Theoremreplimd 10340 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremremimd 10341 Value of the conjugate of a complex number. The value is the real part minus  _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremcjcjd 10342 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.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  ( * `  A ) )  =  A )
 
Theoremreim0bd 10343 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Im `  A )  =  0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrerebd 10344 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Re `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjrebd 10345 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.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( * `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjne0d 10346 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10347 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( * `  A )  =/=  0 )
 
Theoremcjap0d 10347 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( * `  A ) #  0 )
 
Theoremrecjd 10348 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremimcjd 10349 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremcjmulrcld 10350 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  e. 
 RR )
 
Theoremcjmulvald 10351 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremcjmulge0d 10352 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( A  x.  ( * `  A ) ) )
 
Theoremrenegd 10353 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  -u A )  =  -u ( Re `  A ) )
 
Theoremimnegd 10354 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  -u A )  =  -u ( Im `  A ) )
 
Theoremcjnegd 10355 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  -u A )  =  -u ( * `  A ) )
 
Theoremaddcjd 10356 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.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) ) )
 
Theoremcjexpd 10357 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremreaddd 10358 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  +  B ) )  =  ( ( Re
 `  A )  +  ( Re `  B ) ) )
 
Theoremimaddd 10359 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  +  B ) )  =  ( ( Im
 `  A )  +  ( Im `  B ) ) )
 
Theoremresubd 10360 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  -  B ) )  =  ( ( Re
 `  A )  -  ( Re `  B ) ) )
 
Theoremimsubd 10361 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  -  B ) )  =  ( ( Im
 `  A )  -  ( Im `  B ) ) )
 
Theoremremuld 10362 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremimmuld 10363 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im `  A )  x.  ( Re `  B ) ) ) )
 
Theoremcjaddd 10364 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  +  B ) )  =  ( ( * `
  A )  +  ( * `  B ) ) )
 
Theoremcjmuld 10365 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) ) )
 
Theoremipcnd 10366 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremcjdivapd 10367 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B #  0 )   =>    |-  ( ph  ->  ( * `  ( A  /  B ) )  =  ( ( * `  A ) 
 /  ( * `  B ) ) )
 
Theoremrered 10368 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.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Re `  A )  =  A )
 
Theoremreim0d 10369 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Im `  A )  =  0 )
 
Theoremcjred 10370 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( * `  A )  =  A )
 
Theoremremul2d 10371 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( A  x.  ( Re `  B ) ) )
 
Theoremimmul2d 10372 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( A  x.  ( Im `  B ) ) )
 
Theoremredivapd 10373 Real part of a division. Related to remul2 10272. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( Re `  ( B  /  A ) )  =  ( ( Re `  B ) 
 /  A ) )
 
Theoremimdivapd 10374 Imaginary part of a division. Related to remul2 10272. (Contributed by Jim Kingdon, 15-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( Im `  ( B  /  A ) )  =  ( ( Im `  B ) 
 /  A ) )
 
Theoremcrred 10375 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrimd 10376 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
 
3.7.3  Sequence convergence
 
Theoremcaucvgrelemrec 10377* Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( iota_ r  e.  RR  ( A  x.  r
 )  =  1 )  =  ( 1  /  A ) )
 
Theoremcaucvgrelemcau 10378* Lemma for caucvgre 10379. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( F `  n )  <  ( ( F `
  k )  +  ( 1  /  n ) )  /\  ( F `
  k )  < 
 ( ( F `  n )  +  (
 1  /  n )
 ) ) )   =>    |-  ( ph  ->  A. n  e.  NN  A. k  e.  NN  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )
 
Theoremcaucvgre 10379* 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  1  /  n of the nth term.

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

 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( ( F `  n )  <  ( ( F `
  k )  +  ( 1  /  n ) )  /\  ( F `
  k )  < 
 ( ( F `  n )  +  (
 1  /  n )
 ) ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
 ( ( F `  i )  <  ( y  +  x )  /\  y  <  ( ( F `
  i )  +  x ) ) )
 
Theoremcvg1nlemcxze 10380 Lemma for cvg1n 10384. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
 |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  X  e.  RR+ )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  E  e.  NN )   &    |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A )  <  E )   =>    |-  ( ph  ->  ( C  /  ( E  x.  Z ) )  < 
 ( X  /  2
 ) )
 
Theoremcvg1nlemf 10381* Lemma for cvg1n 10384. The modified sequence  G is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  G : NN --> RR )
 
Theoremcvg1nlemcau 10382* Lemma for cvg1n 10384. By selecting spaced out terms for the modified sequence  G, the terms are within  1  /  n (without the constant  C). (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( G `  n )  <  ( ( G `  k )  +  ( 1  /  n ) )  /\  ( G `  k )  <  ( ( G `
  n )  +  ( 1  /  n ) ) ) )
 
Theoremcvg1nlemres 10383* Lemma for cvg1n 10384. The original sequence  F has a limit (turns out it is the same as the limit of the modified sequence  G). (Contributed by Jim Kingdon, 1-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   &    |-  G  =  ( j  e.  NN  |->  ( F `  ( j  x.  Z ) ) )   &    |-  ( ph  ->  Z  e.  NN )   &    |-  ( ph  ->  C  <  Z )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( y  +  x )  /\  y  < 
 ( ( F `  i )  +  x ) ) )
 
Theoremcvg1n 10384* Convergence of real sequences.

This is a version of caucvgre 10379 with a constant multiplier  C on the rate of convergence. That is, all terms after the nth term must be within  C  /  n of the nth term.

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

 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. n  e.  NN  A. k  e.  ( ZZ>= `  n )
 ( ( F `  n )  <  ( ( F `  k )  +  ( C  /  n ) )  /\  ( F `  k )  <  ( ( F `
  n )  +  ( C  /  n ) ) ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( y  +  x )  /\  y  < 
 ( ( F `  i )  +  x ) ) )
 
Theoremuzin2 10385 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( A  e.  ran  ZZ>= 
 /\  B  e.  ran  ZZ>= )  ->  ( A  i^i  B )  e.  ran  ZZ>= )
 
Theoremrexanuz 10386* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
 |-  ( E. j  e. 
 ZZ  A. k  e.  ( ZZ>=
 `  j ) (
 ph  /\  ps )  <->  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph  /\  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ps ) )
 
Theoremrexfiuz 10387* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( A  e.  Fin  ->  ( E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) A. n  e.  A  ph  <->  A. n  e.  A  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph )
 )
 
Theoremrexuz3 10388* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph 
 <-> 
 E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ph )
 )
 
Theoremrexanuz2 10389* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ph  /\  ps )  <->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph  /\  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ps ) )
 
Theoremr19.29uz 10390* A version of 19.29 1556 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( A. k  e.  Z  ph  /\  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ps )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ph  /\  ps )
 )
 
Theoremr19.2uz 10391* A version of r19.2m 3365 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ph  ->  E. k  e.  Z  ph )
 
Theoremrecvguniqlem 10392 Lemma for recvguniq 10393. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  A  <  (
 ( F `  K )  +  ( ( A  -  B )  / 
 2 ) ) )   &    |-  ( ph  ->  ( F `  K )  <  ( B  +  ( ( A  -  B )  / 
 2 ) ) )   =>    |-  ( ph  -> F.  )
 
Theoremrecvguniq 10393* Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) ( ( F `  k
 )  <  ( L  +  x )  /\  L  <  ( ( F `  k )  +  x ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e. 
 NN  A. k  e.  ( ZZ>=
 `  j ) ( ( F `  k
 )  <  ( M  +  x )  /\  M  <  ( ( F `  k )  +  x ) ) )   =>    |-  ( ph  ->  L  =  M )
 
3.7.4  Square root; absolute value
 
Syntaxcsqrt 10394 Extend class notation to include square root of a complex number.
 class  sqr
 
Syntaxcabs 10395 Extend class notation to include a function for the absolute value (modulus) of a complex number.
 class  abs
 
Definitiondf-rsqrt 10396* 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.)

 |- 
 sqr  =  ( x  e.  RR  |->  ( iota_ y  e. 
 RR  ( ( y ^ 2 )  =  x  /\  0  <_  y ) ) )
 
Definitiondf-abs 10397 Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.)
 |- 
 abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
 
Theoremsqrtrval 10398* Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
 |-  ( A  e.  RR  ->  ( sqr `  A )  =  ( iota_ x  e. 
 RR  ( ( x ^ 2 )  =  A  /\  0  <_  x ) ) )
 
Theoremabsval 10399 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.)
 |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
 
Theoremrennim 10400 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( A  e.  RR  ->  ( _i  x.  A )  e/  RR+ )
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