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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreeff1olem 20301* Lemma for reeff1o 20302. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeff1o 20302 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremreefiso 20303 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)

Theoremefcvx 20304 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremreefgim 20305 The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.)
flds        mulGrpflds        GrpIso

13.3.2  Properties of pi = 3.14159...

Theorempilem1 20306 Lemma for pire 20311, pigt2lt4 20309 and sinpi 20310. (Contributed by Mario Carneiro, 9-May-2014.)

Theorempilem2 20307 Lemma for pire 20311, pigt2lt4 20309 and sinpi 20310. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theorempilem3 20308 Lemma for pire 20311, pigt2lt4 20309 and sinpi 20310. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theorempigt2lt4 20309 is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinpi 20310 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempire 20311 is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempipos 20312 is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinhalfpilem 20313 Lemma for sinhalfpi 20315 and coshalfpi 20316. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremhalfpire 20314 is real. (Contributed by David Moews, 28-Feb-2017.)

Theoremsinhalfpi 20315 The sine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcoshalfpi 20316 The cosine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcosneghalfpi 20317 The cosine of is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremefhalfpi 20318 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremcospi 20319 The cosine of is . (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremefipi 20320 The exponential of . (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremeulerid 20321 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsin2pi 20322 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcos2pi 20323 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremef2pi 20324 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremef2kpi 20325 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremefper 20326 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinperlem 20327 Lemma for sinper 20328 and cosper 20329. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsinper 20328 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosper 20329 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2kpi 20330 If is an integer, the sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcos2kpi 20331 If is an integer, the cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2pim 20332 Sine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcos2pim 20333 Cosine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinmpi 20334 Sine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcosmpi 20335 Cosine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinppi 20336 Sine of a number plus . (Contributed by NM, 10-Aug-2008.)

Theoremcosppi 20337 Cosine of a complex number plus . (Contributed by NM, 18-Aug-2008.)

Theoremefimpi 20338 The exponential function of times a real number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinhalfpip 20339 The sine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsinhalfpim 20340 The sine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpip 20341 The cosine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpim 20342 The cosine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremptolemy 20343 Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12714, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. (Contributed by David A. Wheeler, 31-May-2015.)

Theoremsincosq1lem 20344 Lemma for sincosq1sgn 20345. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq1sgn 20345 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq2sgn 20346 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq3sgn 20347 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq4sgn 20348 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoseq00topi 20349 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremcoseq0negpitopi 20350 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremtanrpcl 20351 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtangtx 20352 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanabsge 20353 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremsinq12gt0 20354 The sine of a number strictly between and is positive. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinq12ge0 20355 The sine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsinq34lt0t 20356 The sine of a number strictly between and is negative. (Contributed by NM, 17-Aug-2008.)

Theoremcosq14gt0 20357 The cosine of a number strictly between and is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosq14ge0 20358 The cosine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsincosq1eq 20359 Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos4thpi 20360 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremtan4thpi 20361 The tangent of . (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremsincos6thpi 20362 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos3rdpi 20363 The sine and cosine of . (Contributed by Mario Carneiro, 21-May-2016.)

Theorempige3 20364 is greater or equal to 3. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter . We translate this to algebra by looking at the function as goes from to ; it moves at unit speed and travels distance , hence . (Contributed by Mario Carneiro, 21-May-2016.)

Theoremabssinper 20365 The absolute value of sine has period . (Contributed by NM, 17-Aug-2008.)

Theoremsinkpi 20366 The sine of an integer multiple of is 0. (Contributed by NM, 11-Aug-2008.)

Theoremcoskpi 20367 The absolute value of the cosine of an integer multiple of is 1. (Contributed by NM, 19-Aug-2008.)

Theoremsineq0 20368 A complex number whose sine is zero is an integer multiple of . (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcoseq1 20369 A complex number whose cosine is one is an integer multiple of . (Contributed by Mario Carneiro, 12-May-2014.)

Theoremefeq1 20370 A complex number whose exponential is one is an integer multiple of . (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosne0 20371 The cosine function has no zeroes within the vertical strip of the complex plane between real part and . (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremcosordlem 20372 Lemma for cosord 20373. (Contributed by Mario Carneiro, 10-May-2014.)

Theoremcosord 20373 Cosine is decreasing over the closed interval from to . (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremcos11 20374 Cosine is one-to-one over the closed interval from to . (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinord 20375 Sine is increasing over the closed interval from to . (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremrecosf1o 20376 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremresinf1o 20377 The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremtanord1 20378 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 20379.) (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanord 20379 The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanregt0 20380 The positivity of extends to complex numbers with the same real part. (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremnegpitopissre 20381 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

13.3.3  Mapping of the exponential function

Theoremefgh 20382* The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.)

Theoremefif1olem1 20383* Lemma for efif1o 20387. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem2 20384* Lemma for efif1o 20387. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem3 20385* Lemma for efif1o 20387. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremefif1olem4 20386* The exponential function of an imaginary number maps any interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Theoremefif1o 20387* The exponential function of an imaginary number maps any open-below, closed-above interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremefifo 20388* The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1olem 20389* The exponential function maps the set , of complex numbers with imaginary part in a real interval of length , one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Theoremeff1o 20390 The exponential function maps the set , of complex numbers with imaginary part in the closed-above, open-below interval from to one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

13.3.4  The natural logarithm on complex numbers

Syntaxclog 20391 Extend class notation with the natural logarithm function on complex numbers.

Syntaxccxp 20392 Extend class notation with the complex power function.

Definitiondf-log 20393 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). (Contributed by Paul Chapman, 21-Apr-2008.)

Definitiondf-cxp 20394* Define the power function on complex numbers. Note that the value of this function when and should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogrn 20395 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremellogrn 20396 Write out the property explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremdflog2 20397 The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogrn 20398 The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)

Theoremlogrncn 20399 The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1o2 20400 The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

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