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Type | Label | Description |
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Statement | ||
Theorem | sgnp 13201 | Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.) |
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Theorem | sgnrrp 13202 | Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.) |
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Theorem | sgn1 13203 | Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.) |
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Theorem | sgnpnf 13204 |
Proof that the signum of ![]() |
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Theorem | sgnn 13205 | Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.) |
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Theorem | sgnmnf 13206 |
Proof that the signum of ![]() |
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Syntax | ccj 13207 | Extend class notation to include complex conjugate function. |
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Syntax | cre 13208 | Extend class notation to include real part of a complex number. |
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Syntax | cim 13209 | Extend class notation to include imaginary part of a complex number. |
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Definition | df-cj 13210* | Define the complex conjugate function. See cjcli 13280 for its closure and cjval 13213 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Definition | df-re 13211 | Define a function whose value is the real part of a complex number. See reval 13217 for its value, recli 13278 for its closure, and replim 13227 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
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Definition | df-im 13212 | Define a function whose value is the imaginary part of a complex number. See imval 13218 for its value, imcli 13279 for its closure, and replim 13227 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
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Theorem | cjval 13213* | The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.) |
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Theorem | cjth 13214 | The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.) |
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Theorem | cjf 13215 | Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.) |
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Theorem | cjcl 13216 | 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.) |
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Theorem | reval 13217 | The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | imval 13218 | The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | imre 13219 | 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.) |
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Theorem | reim 13220 | The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
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Theorem | recl 13221 | The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | imcl 13222 | The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | ref 13223 | Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | imf 13224 | Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
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Theorem | crre 13225 | 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.) |
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Theorem | crim 13226 | 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.) |
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Theorem | replim 13227 | Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
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Theorem | remim 13228 |
Value of the conjugate of a complex number. The value is the real part
minus ![]() |
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Theorem | reim0 13229 | The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
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Theorem | reim0b 13230 | A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.) |
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Theorem | rereb 13231 | 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.) |
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Theorem | mulre 13232 | A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.) |
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Theorem | rere 13233 | 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.) |
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Theorem | cjreb 13234 | 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.) |
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Theorem | recj 13235 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.) |
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Theorem | reneg 13236 | Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | readd 13237 | Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | resub 13238 | Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.) |
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Theorem | remullem 13239 | Lemma for remul 13240, immul 13247, and cjmul 13253. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | remul 13240 | Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | remul2 13241 | Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | rediv 13242 | Real part of a division. Related to remul2 13241. (Contributed by David A. Wheeler, 10-Jun-2015.) |
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Theorem | imcj 13243 | Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | imneg 13244 | The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | imadd 13245 | Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | imsub 13246 | Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.) |
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Theorem | immul 13247 | Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | immul2 13248 | Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
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Theorem | imdiv 13249 | Imaginary part of a division. Related to immul2 13248. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | cjre 13250 | A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.) |
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Theorem | cjcj 13251 | 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.) |
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Theorem | cjadd 13252 | 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.) |
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Theorem | cjmul 13253 | 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.) |
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Theorem | ipcnval 13254 | Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | cjmulrcl 13255 | A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | cjmulval 13256 | A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | cjmulge0 13257 | A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | cjneg 13258 | Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
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Theorem | addcj 13259 | 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.) |
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Theorem | cjsub 13260 | Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.) |
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Theorem | cjexp 13261 | Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.) |
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Theorem | imval2 13262 | The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.) |
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Theorem | re0 13263 | The real part of zero. (Contributed by NM, 27-Jul-1999.) |
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Theorem | im0 13264 | The imaginary part of zero. (Contributed by NM, 27-Jul-1999.) |
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Theorem | re1 13265 | The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
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Theorem | im1 13266 | The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.) |
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Theorem | rei 13267 |
The real part of ![]() |
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Theorem | imi 13268 |
The imaginary part of ![]() |
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Theorem | cj0 13269 | The conjugate of zero. (Contributed by NM, 27-Jul-1999.) |
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Theorem | cji 13270 | The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
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Theorem | cjreim 13271 | The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) |
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Theorem | cjreim2 13272 | 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.) |
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Theorem | cj11 13273 | 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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Theorem | cjne0 13274 | A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.) |
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Theorem | cjdiv 13275 | Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | cnrecnv 13276* |
The inverse to the canonical bijection from ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sqeqd 13277 | A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.) |
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Theorem | recli 13278 | The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
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Theorem | imcli 13279 | The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.) |
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Theorem | cjcli 13280 | Closure law for complex conjugate. (Contributed by NM, 11-May-1999.) |
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Theorem | replimi 13281 | Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.) |
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Theorem | cjcji 13282 | 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.) |
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Theorem | reim0bi 13283 | A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.) |
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Theorem | rerebi 13284 | A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.) |
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Theorem | cjrebi 13285 | 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.) |
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Theorem | recji 13286 | Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | imcji 13287 | Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjmulrcli 13288 | A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.) |
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Theorem | cjmulvali 13289 | A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjmulge0i 13290 | A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.) |
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Theorem | renegi 13291 | Real part of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | imnegi 13292 | Imaginary part of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | cjnegi 13293 | Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.) |
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Theorem | addcji 13294 | 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.) |
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Theorem | readdi 13295 | Real part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
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Theorem | imaddi 13296 | Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.) |
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Theorem | remuli 13297 | Real part of a product. (Contributed by NM, 28-Jul-1999.) |
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Theorem | immuli 13298 | Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) |
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Theorem | cjaddi 13299 | Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) |
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Theorem | cjmuli 13300 | Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.) |
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