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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremremul2 9101 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 RR  CC  Re `  x.  x.  Re `
 
Theoremredivap 9102 Real part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  RR #  0  Re `  Re `
 
Theoremimcj 9103 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  Im `  * `  -u Im `
 
Theoremimneg 9104 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  Im `  -u  -u Im
 `
 
Theoremimadd 9105 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Im `  +  Im `  +  Im `
 
Theoremimsub 9106 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
 CC  CC  Im `  -  Im ` 
 -  Im `
 
Theoremimmul 9107 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Im `  x.  Re `  x.  Im `  +  Im `  x.  Re `
 
Theoremimmul2 9108 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 RR  CC  Im `  x.  x.  Im `
 
Theoremimdivap 9109 Imaginary part of a division. Related to immul2 9108. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  RR #  0  Im `  Im `
 
Theoremcjre 9110 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
 RR  * `
 
Theoremcjcj 9111 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.)
 CC  * `  * `
 
Theoremcjadd 9112 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.)
 CC  CC  * `  +  * `  +  * `
 
Theoremcjmul 9113 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.)
 CC  CC  * `  x.  * `  x.  * `
 
Theoremipcnval 9114 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  CC  Re `  x.  * `  Re `  x.  Re `  +  Im `  x.  Im `
 
Theoremcjmulrcl 9115 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  x.  * `  RR
 
Theoremcjmulval 9116 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  x.  * `  Re ` 
 ^ 2  +  Im `  ^ 2
 
Theoremcjmulge0 9117 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  0  <_  x.  * `
 
Theoremcjneg 9118 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  * `  -u  -u * `
 
Theoremaddcj 9119 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.)
 CC  +  * `  2  x.  Re `
 
Theoremcjsub 9120 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 CC  CC  * `  -  * ` 
 -  * `
 
Theoremcjexp 9121 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
 CC  N  NN0  * `  ^ N  * ` 
 ^ N
 
Theoremimval2 9122 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
 CC  Im `  -  * ` 
 2  x.  _i
 
Theoremre0 9123 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 Re `  0  0
 
Theoremim0 9124 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 Im `  0  0
 
Theoremre1 9125 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 Re `  1  1
 
Theoremim1 9126 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 Im `  1  0
 
Theoremrei 9127 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 Re `  _i  0
 
Theoremimi 9128 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 Im `  _i  1
 
Theoremcj0 9129 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 * `  0  0
 
Theoremcji 9130 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 * `  _i  -u _i
 
Theoremcjreim 9131 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
 RR  RR  * `  +  _i  x. 
 -  _i  x.
 
Theoremcjreim2 9132 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.)
 RR  RR  * `  -  _i  x.  +  _i  x.
 
Theoremcj11 9133 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
 CC  CC  * `
  * `
 
Theoremcjap 9134 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  CC  * `
 #  * ` #
 
Theoremcjap0 9135 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC #  0  * `
 #  0
 
Theoremcjne0 9136 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
 CC  =/=  0  * `  =/=  0
 
Theoremcjdivap 9137 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC  CC #  0  * `  * `  * `
 
Theoremcnrecnv 9138* The inverse to the canonical bijection from  RR  X.  RR to  CC from cnref1o 8357. (Contributed by Mario Carneiro, 25-Aug-2014.)
 F  RR ,  RR  |->  +  _i  x.    =>     `' F  CC  |->  <. Re
 `  ,  Im ` 
 >.
 
Theoremrecli 9139 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 CC   =>     Re `  RR
 
Theoremimcli 9140 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 CC   =>     Im `  RR
 
Theoremcjcli 9141 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 CC   =>     * `  CC
 
Theoremreplimi 9142 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
 CC   =>     Re `  +  _i  x.  Im `
 
Theoremcjcji 9143 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.)
 CC   =>     * `  * `
 
Theoremreim0bi 9144 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
 CC   =>     RR  Im `  0
 
Theoremrerebi 9145 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
 CC   =>     RR  Re `
 
Theoremcjrebi 9146 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.)
 CC   =>     RR  * `
 
Theoremrecji 9147 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 CC   =>     Re `  * `  Re `
 
Theoremimcji 9148 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 CC   =>     Im `  * `  -u Im `
 
Theoremcjmulrcli 9149 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 CC   =>     x.  * `  RR
 
Theoremcjmulvali 9150 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
 CC   =>     x.  * `  Re ` 
 ^ 2  +  Im `  ^ 2
 
Theoremcjmulge0i 9151 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 CC   =>     0  <_  x.  * `
 
Theoremrenegi 9152 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 CC   =>     Re `  -u  -u Re
 `
 
Theoremimnegi 9153 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 CC   =>     Im `  -u  -u Im
 `
 
Theoremcjnegi 9154 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 CC   =>     * `  -u  -u * `
 
Theoremaddcji 9155 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.)
 CC   =>     +  * `  2  x.  Re `
 
Theoremreaddi 9156 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     Re `  +  Re
 `  +  Re `
 
Theoremimaddi 9157 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     Im `  +  Im
 `  +  Im `
 
Theoremremuli 9158 Real part of a product. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     Re `  x.  Re `  x.  Re `  -  Im `  x.  Im `
 
Theoremimmuli 9159 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     Im `  x.  Re `  x.  Im `  +  Im `  x.  Re `
 
Theoremcjaddi 9160 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     * `  +  * `
  +  * `
 
Theoremcjmuli 9161 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 CC   &     CC   =>     * `  x.  * `
  x.  * `
 
Theoremipcni 9162 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
 CC   &     CC   =>     Re `  x.  * `  Re
 `  x.  Re `  +  Im `  x.  Im `
 
Theoremcjdivapi 9163 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 CC   &     CC   =>    #  0  * `  * `  * `
 
Theoremcrrei 9164 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 RR   &     RR   =>     Re `  +  _i  x.
 
Theoremcrimi 9165 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 RR   &     RR   =>     Im `  +  _i  x.
 
Theoremrecld 9166 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Re ` 
 RR
 
Theoremimcld 9167 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Im ` 
 RR
 
Theoremcjcld 9168 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     * ` 
 CC
 
Theoremreplimd 9169 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Re `  +  _i  x.  Im `
 
Theoremremimd 9170 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.)
 CC   =>     * `  Re `  -  _i  x.  Im `
 
Theoremcjcjd 9171 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.)
 CC   =>     * `  * `
 
Theoremreim0bd 9172 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     Im `  0   =>     RR
 
Theoremrerebd 9173 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     Re `    =>     RR
 
Theoremcjrebd 9174 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.)
 CC   &     * `    =>     RR
 
Theoremcjne0d 9175 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     =/=  0   =>     * `  =/=  0
 
Theoremrecjd 9176 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Re `  * `  Re `
 
Theoremimcjd 9177 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Im `  * `  -u Im `
 
Theoremcjmulrcld 9178 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     x.  * `  RR
 
Theoremcjmulvald 9179 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     x.  * `  Re `  ^ 2  +  Im
 `  ^
 2
 
Theoremcjmulge0d 9180 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     0  <_  x.  * `
 
Theoremrenegd 9181 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Re `  -u  -u Re `
 
Theoremimnegd 9182 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     Im `  -u  -u Im `
 
Theoremcjnegd 9183 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   =>     * `  -u  -u * `
 
Theoremaddcjd 9184 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.)
 CC   =>     +  * `  2  x.  Re
 `
 
Theoremcjexpd 9185 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     N  NN0   =>     * `  ^ N  * `  ^ N
 
Theoremreaddd 9186 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Re `  +  Re `  +  Re `
 
Theoremimaddd 9187 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Im `  +  Im `  +  Im `
 
Theoremresubd 9188 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Re `  -  Re `  -  Re `
 
Theoremimsubd 9189 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Im `  -  Im `  -  Im `
 
Theoremremuld 9190 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Re `  x.  Re `  x.  Re ` 
 -  Im
 `  x.  Im `
 
Theoremimmuld 9191 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Im `  x.  Re `  x.  Im `  +  Im
 `  x.  Re `
 
Theoremcjaddd 9192 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     * `  +  * `  +  * `
 
Theoremcjmuld 9193 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     * `  x.  * `  x.  * `
 
Theoremipcnd 9194 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
 CC   &     CC   =>     Re `  x.  * `  Re `  x.  Re `  +  Im
 `  x.  Im `
 
Theoremcjdivapd 9195 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
 CC   &     CC   &    #  0   =>     * `  * `  * `
 
Theoremrered 9196 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.)
 RR   =>     Re `
 
Theoremreim0d 9197 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 RR   =>     Im `  0
 
Theoremcjred 9198 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 RR   =>     * `
 
Theoremremul2d 9199 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 RR   &     CC   =>     Re `  x.  x.  Re
 `
 
Theoremimmul2d 9200 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 RR   &     CC   =>     Im `  x.  x.  Im
 `
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