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Type | Label | Description |
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Statement | ||
Theorem | ipcni 13301 | Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.) |
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Theorem | cjdivi 13302 | Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | crrei 13303 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
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Theorem | crimi 13304 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) |
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Theorem | recld 13305 | The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imcld 13306 | The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjcld 13307 | Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | replimd 13308 | Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remimd 13309 |
Value of the conjugate of a complex number. The value is the real part
minus ![]() |
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Theorem | cjcjd 13310 | 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.) |
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Theorem | reim0bd 13311 | A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | rerebd 13312 | A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjrebd 13313 | 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.) |
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Theorem | cjne0d 13314 | A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | recjd 13315 | Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imcjd 13316 | Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulrcld 13317 | A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulvald 13318 | A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmulge0d 13319 | A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | renegd 13320 | Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imnegd 13321 | Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjnegd 13322 | Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | addcjd 13323 | 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.) |
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Theorem | cjexpd 13324 | Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | readdd 13325 | Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imaddd 13326 | Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | resubd 13327 | Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imsubd 13328 | Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remuld 13329 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immuld 13330 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjaddd 13331 | Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjmuld 13332 | Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | ipcnd 13333 | Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjdivd 13334 | Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | rered 13335 | 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.) |
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Theorem | reim0d 13336 | The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | cjred 13337 | A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | remul2d 13338 | Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | immul2d 13339 | Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | redivd 13340 | Real part of a division. Related to remul2 13241. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | imdivd 13341 | Imaginary part of a division. Related to remul2 13241. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | crred 13342 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
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Theorem | crimd 13343 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
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Syntax | csqrt 13344 | Extend class notation to include square root of a complex number. |
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Syntax | cabs 13345 | Extend class notation to include a function for the absolute value (modulus) of a complex number. |
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Definition | df-sqrt 13346* |
Define a function whose value is the square root of a complex number.
Since ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() See sqrtcl 13472 for its closure, sqrtval 13348 for its value, sqrtth 13475 and sqsqrti 13486 for its relationship to squares, and sqrt11i 13495 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.) |
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Definition | df-abs 13347 | Define the function for the absolute value (modulus) of a complex number. See abscli 13505 for its closure and absval 13349 or absval2i 13507 for its value. (Contributed by NM, 27-Jul-1999.) |
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Theorem | sqrtval 13348* | Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.) |
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Theorem | absval 13349 | 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.) |
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Theorem | rennim 13350 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
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Theorem | cnpart 13351 |
The specification of restriction to the right half-plane partitions the
complex plane without 0 into two disjoint pieces, which are related by a
reflection about the origin (under the map ![]() ![]() ![]() ![]() |
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Theorem | sqr0lem 13352 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrt0 13353 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrlem1 13354* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem2 13355* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem3 13356* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem4 13357* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem5 13358* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem6 13359* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrlem7 13360* | Lemma for 01sqrex 13361. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | 01sqrex 13361* |
Existence of a square root for reals in the interval ![]() ![]() ![]() ![]() ![]() |
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Theorem | resqrex 13362* | Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrmo 13363* | Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.) |
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Theorem | resqreu 13364* | Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | resqrtcl 13365 | Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | resqrtthlem 13366 | Lemma for resqrtth 13367. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | resqrtth 13367 | Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | remsqsqrt 13368 | Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.) |
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Theorem | sqrtge0 13369 | The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrtgt0 13370 | The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.) |
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Theorem | sqrtmul 13371 | Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtle 13372 | Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtlt 13373 | Square root is strictly monotonic. Closed form of sqrtlti 13500. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrt11 13374 | The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.) |
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Theorem | sqrt00 13375 | A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | rpsqrtcl 13376 | The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.) |
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Theorem | sqrtdiv 13377 | Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.) |
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Theorem | sqrtneglem 13378 | The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrtneg 13379 | The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.) |
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Theorem | sqrtsq2 13380 | Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtsq 13381 | Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrtmsq 13382 | Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | sqrt1 13383 | The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.) |
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Theorem | sqrt4 13384 | The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.) |
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Theorem | sqrt9 13385 | The square root of 9 is 3. (Contributed by NM, 11-May-2004.) |
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Theorem | sqrt2gt1lt2 13386 | The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.) |
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Theorem | sqrtm1 13387 |
The imaginary unit is the square root of negative 1. A lot of people like
to call this the "definition" of ![]() ![]() ![]() |
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Theorem | absneg 13388 | Absolute value of negative. (Contributed by NM, 27-Feb-2005.) |
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Theorem | abscl 13389 | Real closure of absolute value. (Contributed by NM, 3-Oct-1999.) |
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Theorem | abscj 13390 | The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.) |
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Theorem | absvalsq 13391 | Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.) |
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Theorem | absvalsq2 13392 | Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.) |
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Theorem | sqabsadd 13393 | Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) |
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Theorem | sqabssub 13394 | Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.) |
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Theorem | absval2 13395 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.) |
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Theorem | abs0 13396 | The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | absi 13397 | The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.) |
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Theorem | absge0 13398 | Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.) |
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Theorem | absrpcl 13399 | The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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Theorem | abs00 13400 | The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
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