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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcosneg 12301 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)

Theoremtanneg 12302 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)

Theoremsin0 12303 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)

Theoremcos0 12304 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)

Theoremtan0 12305 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)

Theoremefival 12306 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)

Theoremefmival 12307 The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)

Theoremsinhval 12308 Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremcoshval 12309 Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremresinhcl 12310 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremrpcoshcl 12311 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremrecoshcl 12312 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremretanhcl 12313 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhlt1 12314 The hyperbolic tangent of a real number is upper bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhbnd 12315 The hyperbolic tangent of a real number is bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremefeul 12316 Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)

Theoremefieq 12317 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinadd 12318 Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcosadd 12319 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremtanaddlem 12320 A useful intermediate step in tanadd 12321 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremsinsub 12322 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremcossub 12323 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremaddsin 12324 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubsin 12325 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsinmul 12326 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12319 and cossub 12323. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremcosmul 12327 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12319 and cossub 12323. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremaddcos 12328 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubcos 12329 Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsincossq 12330 Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.)

Theoremsin2t 12331 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremcos2t 12332 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcos2tsin 12333 Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)

Theoremsinbnd 12334 The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremcosbnd 12335 The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremsinbnd2 12336 The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremcosbnd2 12337 The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremef01bndlem 12338* Lemma for sin01bnd 12339 and cos01bnd 12340. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin01bnd 12339 Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcos01bnd 12340 Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcos1bnd 12341 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos2bnd 12342 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsinltx 12343 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremsin01gt0 12344 The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos01gt0 12345 The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin02gt0 12346 The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos1sgn 12347 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos2sgn 12348 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin4lt0 12349 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremabsefi 12350 The absolute value of the exponential function of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)

Theoremabsef 12351 The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)

Theoremabsefib 12352 A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.)

Theoremefieq1re 12353 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)

Theoremdemoivre 12354 De Moivre's Formula. Shorter proof of demoivreALT 12355 using the exponential function. (Contributed by NM, 24-Jul-2007.)

TheoremdemoivreALT 12355 De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.)

5.9.2  _e is irrational

Theoremeirrlem 12356* Lemma for eirr 12357. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeirr 12357 is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremegt2lt3 12358 Euler's constant = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremepos 12359 Euler's constant is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)

Theoremepr 12360 Euler's constant is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)

5.10  Cardinality of real and complex number subsets

5.10.1  Countability of integers and rationals

Theoremxpnnen 12361 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

TheoremxpnnenOLD 12362 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. The key idea is to use nn0opth2 11165 to show that the mapping from natural numbers and to is one-to-one. (Contributed by NM, 1-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremxpomenOLD 12363 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133 (which proves this with a direct, but longer, proof; ours uses instead the Schroeder-Bernstein Theorem sbth 6866 in xpnnen 12361). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremznnenlem 12364 Lemma for znnen 12365. (Contributed by NM, 31-Jul-2004.)

Theoremznnen 12365 The set of integers and the set of natural numbers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)

Theoremqnnen 12366 The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set is numerable. Exercise 2 of [Enderton] p. 133. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.)

5.10.2  The reals are uncountable

Theoremrpnnen2lem1 12367* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem2 12368* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremrpnnen2lem3 12369* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem4 12370* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)

Theoremrpnnen2lem5 12371* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem6 12372* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem7 12373* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem8 12374* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem9 12375* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem10 12376* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem11 12377* Lemma for rpnnen2 12378. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2 12378* The other half of rpnnen 12379, where we show an injection from sets of natural numbers to real numbers. The obvious choice for this is binary expansion, but it has the unfortunate property that it does not produce an injection on numbers which end with all 0's or all 1's (the more well-known decimal version of this is 0.999... 12211). Instead, we opt for a ternary expansion, which produces (a scaled version of) the Cantor set. Since the Cantor set is riddled with gaps, we can show that any two sequences that are not equal must differ somewhere, and when they do, they are placed a finite distance apart, thus ensuring that the map is injective.

Our map assigns to each subset of the natural numbers the number , where (rpnnen2lem1 12367). This is an infinite sum of real numbers (rpnnen2lem2 12368), and since implies (rpnnen2lem4 12370) and converges to (rpnnen2lem3 12369) by geoisum1 12209, the sum is convergent to some real (rpnnen2lem5 12371 and rpnnen2lem6 12372) by the comparison test for convergence cvgcmp 12151. The comparison test also tells us that implies (rpnnen2lem7 12373).

Putting it all together, if we have two sets , there must differ somewhere, and so there must be an such that but or vice versa. In this case, we split off the first terms (rpnnen2lem8 12374) and cancel them (rpnnen2lem10 12376), since these are the same for both sets. For the remaining terms, we use the subset property to establish that and (where these sums are only over ), and since (rpnnen2lem9 12375) and , we establish that (rpnnen2lem11 12377) so that they must be different. By contraposition, we find that this map is an injection. (Contributed by Mario Carneiro, 13-May-2013.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen 12379 The cardinality of the continuum is the same as the powerset of . This is a stronger statement than ruc 12395, which only asserts that is uncountable, i.e. has a cardinality larger than . The main proof is in two parts, rpnnen1 10226 and rpnnen2 12378, each showing an injection in one direction, and this last part uses sbth 6866 to prove that the sets are equinumerous. By constructing explicit injections, we avoid the use of AC. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremrexpen 12380 The real numbers are equinumerous to their own cross product, even though it is not necessarily true that is well-orderable (so we cannot use infxpidm2 7528 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremcpnnen 12381 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)

TheoremrucALT 12382 The set of natural numbers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. This proof is a simple corollary of rpnnen 12379, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpegif/mmcomplex.html#uncountable, see ruc 12395. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.)

Theoremruclem1 12383* Lemma for ruc 12395 (the reals are uncountable). Substitutions for the function . (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

Theoremruclem2 12384* Lemma for ruc 12395. Ordering property for the input to . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem3 12385* Lemma for ruc 12395. The constructed interval always excludes . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem4 12386* Lemma for ruc 12395. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem6 12387* Lemma for ruc 12395. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem7 12388* Lemma for ruc 12395. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem8 12389* Lemma for ruc 12395. The intervals of the sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem9 12390* Lemma for ruc 12395. The first components of the sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem10 12391* Lemma for ruc 12395. Every first component of the sequence is less than every second component. That is, the sequences form a chain a1 a2 ... b2 b1, where ai are the first components and bi are the second components. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem11 12392* Lemma for ruc 12395. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem12 12393* Lemma for ruc 12395. The supremum of the increasing sequence is a real number that is not in the range of . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem13 12394 Lemma for ruc 12395. There is no function that maps onto . (Use nex 1587 if you want this in the form .) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Theoremruc 12395 The set of natural numbers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 12383 through ruclem13 12394 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12394 for the function existence version of this theorem. For an informal discussion of this proof, see http://us.metamath.org/mpegif/mmcomplex.html#uncountable. For an alternate proof see rucALT 12382. (Contributed by NM, 13-Oct-2004.) (Proof modification is discouraged.)

Theoremresdomq 12396 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)

Theoremaleph1re 12397 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)

Theoremaleph1irr 12398 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)

Theoremcnso 12399 The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.)

PART 6  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

6.1  Elementary properties of divisibility

6.1.1  Irrationality of square root of 2

Theoremsqr2irrlem 12400 Lemma for irrationality of square root of 2. The core of the proof - if , then and are even, so and are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

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