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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremretancld 12701 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsinneg 12702 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)

Theoremcosneg 12703 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)

Theoremtanneg 12704 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)

Theoremsin0 12705 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)

Theoremcos0 12706 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)

Theoremtan0 12707 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)

Theoremefival 12708 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)

Theoremefmival 12709 The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)

Theoremsinhval 12710 Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremcoshval 12711 Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremresinhcl 12712 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremrpcoshcl 12713 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremrecoshcl 12714 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremretanhcl 12715 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhlt1 12716 The hyperbolic tangent of a real number is upper bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhbnd 12717 The hyperbolic tangent of a real number is bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremefeul 12718 Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)

Theoremefieq 12719 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinadd 12720 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 12721 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremtanaddlem 12722 A useful intermediate step in tanadd 12723 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremsinsub 12724 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremcossub 12725 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremaddsin 12726 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubsin 12727 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsinmul 12728 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12721 and cossub 12725. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremcosmul 12729 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12721 and cossub 12725. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremaddcos 12730 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubcos 12731 Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsincossq 12732 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 12733 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremcos2t 12734 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcos2tsin 12735 Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)

Theoremsinbnd 12736 The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremcosbnd 12737 The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremsinbnd2 12738 The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremcosbnd2 12739 The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremef01bndlem 12740* Lemma for sin01bnd 12741 and cos01bnd 12742. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin01bnd 12741 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 12742 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 12743 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos2bnd 12744 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsinltx 12745 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremsin01gt0 12746 The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos01gt0 12747 The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin02gt0 12748 The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos1sgn 12749 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos2sgn 12750 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin4lt0 12751 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremabsefi 12752 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 12753 The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)

Theoremabsefib 12754 A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.)

Theoremefieq1re 12755 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)

Theoremdemoivre 12756 De Moivre's Formula. Shorter proof of demoivreALT 12757 using the exponential function. (Contributed by NM, 24-Jul-2007.)

TheoremdemoivreALT 12757 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.) (New usage is discouraged.)

5.9.2  _e is irrational

Theoremeirrlem 12758* Lemma for eirr 12759. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeirr 12759 is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremegt2lt3 12760 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 12761 Euler's constant is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)

Theoremepr 12762 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 12763 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 12764 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 11520 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 12765 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 7186 in xpnnen 12763). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremznnenlem 12766 Lemma for znnen 12767. (Contributed by NM, 31-Jul-2004.)

Theoremznnen 12767 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 12768 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 12769* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem2 12770* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremrpnnen2lem3 12771* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem4 12772* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)

Theoremrpnnen2lem5 12773* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem6 12774* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem7 12775* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem8 12776* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem9 12777* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem10 12778* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem11 12779* Lemma for rpnnen2 12780. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2 12780* The other half of rpnnen 12781, 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... 12613). 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 12769). This is an infinite sum of real numbers (rpnnen2lem2 12770), and since implies (rpnnen2lem4 12772) and converges to (rpnnen2lem3 12771) by geoisum1 12611, the sum is convergent to some real (rpnnen2lem5 12773 and rpnnen2lem6 12774) by the comparison test for convergence cvgcmp 12550. The comparison test also tells us that implies (rpnnen2lem7 12775).

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 12776) and cancel them (rpnnen2lem10 12778), 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 12777) and , we establish that (rpnnen2lem11 12779) 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 12781 The cardinality of the continuum is the same as the powerset of . This is a stronger statement than ruc 12797, which only asserts that is uncountable, i.e. has a cardinality larger than . The main proof is in two parts, rpnnen1 10561 and rpnnen2 12780, each showing an injection in one direction, and this last part uses sbth 7186 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 12782 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 7854 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremcpnnen 12783 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)

TheoremrucALT 12784 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 12781, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpeuni/mmcomplex.html#uncountable, see ruc 12797. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremruclem1 12785* Lemma for ruc 12797 (the reals are uncountable). Substitutions for the function . (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

Theoremruclem2 12786* Lemma for ruc 12797. Ordering property for the input to . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem3 12787* Lemma for ruc 12797. The constructed interval always excludes . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem4 12788* Lemma for ruc 12797. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem6 12789* Lemma for ruc 12797. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem7 12790* Lemma for ruc 12797. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem8 12791* Lemma for ruc 12797. The intervals of the sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem9 12792* Lemma for ruc 12797. The first components of the sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem10 12793* Lemma for ruc 12797. 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 12794* Lemma for ruc 12797. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem12 12795* Lemma for ruc 12797. 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 12796 Lemma for ruc 12797. There is no function that maps onto . (Use nex 1561 if you want this in the form .) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Theoremruc 12797 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 12785 through ruclem13 12796 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12796 for the function existence version of this theorem. For an informal discussion of this proof, see http://us.metamath.org/mpeuni/mmcomplex.html#uncountable. For an alternate proof see rucALT 12784. (Contributed by NM, 13-Oct-2004.)

Theoremresdomq 12798 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)

Theoremaleph1re 12799 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)

Theoremaleph1irr 12800 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)

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