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

Theoremabelthlem2 20301* Lemma for abelth 20310. The peculiar region , known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing . Indeed, except for itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem3 20302* Lemma for abelth 20310. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem4 20303* Lemma for abelth 20310. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem5 20304* Lemma for abelth 20310. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelthlem6 20305* Lemma for abelth 20310. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem7a 20306* Lemma for abelth 20310. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremabelthlem7 20307* Lemma for abelth 20310. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem8 20308* Lemma for abelth 20310. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem9 20309* Lemma for abelth 20310. By adjusting the constant term, we can assume that the entire series converges to . (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelth 20310* Abel's theorem. If the power series is convergent at , then it is equal to the limit from "below", along a Stolz angle (note that the case of a Stolz angle is the real line ). (Continuity on follows more generally from psercn 20295.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremabelth2 20311* Abel's theorem, restricted to the interval. (Contributed by Mario Carneiro, 2-Apr-2015.)

13.3  Basic trigonometry

13.3.1  The exponential, sine, and cosine functions (cont.)

Theoremefcn 20312 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theoremsincn 20313 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremcoscn 20314 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremreeff1olem 20315* Lemma for reeff1o 20316. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeff1o 20316 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremreefiso 20317 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 20318 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremreefgim 20319 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 20320 Lemma for pire 20325, pigt2lt4 20323 and sinpi 20324. (Contributed by Mario Carneiro, 9-May-2014.)

Theorempilem2 20321 Lemma for pire 20325, pigt2lt4 20323 and sinpi 20324. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theorempilem3 20322 Lemma for pire 20325, pigt2lt4 20323 and sinpi 20324. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theorempigt2lt4 20323 is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinpi 20324 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempire 20325 is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempipos 20326 is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinhalfpilem 20327 Lemma for sinhalfpi 20329 and coshalfpi 20330. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremhalfpire 20328 is real. (Contributed by David Moews, 28-Feb-2017.)

Theoremsinhalfpi 20329 The sine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcoshalfpi 20330 The cosine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcosneghalfpi 20331 The cosine of is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremefhalfpi 20332 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremcospi 20333 The cosine of is . (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremefipi 20334 The exponential of . (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremeulerid 20335 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsin2pi 20336 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcos2pi 20337 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremef2pi 20338 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremef2kpi 20339 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremefper 20340 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinperlem 20341 Lemma for sinper 20342 and cosper 20343. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsinper 20342 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosper 20343 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2kpi 20344 If is an integer, the sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcos2kpi 20345 If is an integer, the cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2pim 20346 Sine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcos2pim 20347 Cosine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinmpi 20348 Sine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcosmpi 20349 Cosine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinppi 20350 Sine of a number plus . (Contributed by NM, 10-Aug-2008.)

Theoremcosppi 20351 Cosine of a complex number plus . (Contributed by NM, 18-Aug-2008.)

Theoremefimpi 20352 The exponential function of times a real number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinhalfpip 20353 The sine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsinhalfpim 20354 The sine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpip 20355 The cosine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpim 20356 The cosine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremptolemy 20357 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 12728, 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 20358 Lemma for sincosq1sgn 20359. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq1sgn 20359 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq2sgn 20360 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq3sgn 20361 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq4sgn 20362 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoseq00topi 20363 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremcoseq0negpitopi 20364 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremtanrpcl 20365 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtangtx 20366 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanabsge 20367 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremsinq12gt0 20368 The sine of a number strictly between and is positive. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinq12ge0 20369 The sine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsinq34lt0t 20370 The sine of a number strictly between and is negative. (Contributed by NM, 17-Aug-2008.)

Theoremcosq14gt0 20371 The cosine of a number strictly between and is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosq14ge0 20372 The cosine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsincosq1eq 20373 Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos4thpi 20374 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremtan4thpi 20375 The tangent of . (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremsincos6thpi 20376 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos3rdpi 20377 The sine and cosine of . (Contributed by Mario Carneiro, 21-May-2016.)

Theorempige3 20378 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 20379 The absolute value of sine has period . (Contributed by NM, 17-Aug-2008.)

Theoremsinkpi 20380 The sine of an integer multiple of is 0. (Contributed by NM, 11-Aug-2008.)

Theoremcoskpi 20381 The absolute value of the cosine of an integer multiple of is 1. (Contributed by NM, 19-Aug-2008.)

Theoremsineq0 20382 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 20383 A complex number whose cosine is one is an integer multiple of . (Contributed by Mario Carneiro, 12-May-2014.)

Theoremefeq1 20384 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 20385 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 20386 Lemma for cosord 20387. (Contributed by Mario Carneiro, 10-May-2014.)

Theoremcosord 20387 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 20388 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 20389 Sine is increasing over the closed interval from to . (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremrecosf1o 20390 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremresinf1o 20391 The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremtanord1 20392 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 20393.) (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanord 20393 The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanregt0 20394 The positivity of extends to complex numbers with the same real part. (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremnegpitopissre 20395 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

13.3.3  Mapping of the exponential function

Theoremefgh 20396* 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 20397* Lemma for efif1o 20401. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem2 20398* Lemma for efif1o 20401. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem3 20399* Lemma for efif1o 20401. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremefif1olem4 20400* 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.)

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