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

Theoremretanhcl 12701 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhlt1 12702 The hyperbolic tangent of a real number is upper bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanhbnd 12703 The hyperbolic tangent of a real number is bounded by . (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremefeul 12704 Eulerian representation of the complex exponential. (Suggested by Jeffrey Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)

Theoremefieq 12705 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinadd 12706 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 12707 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremtanaddlem 12708 A useful intermediate step in tanadd 12709 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremsinsub 12710 Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremcossub 12711 Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremaddsin 12712 Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubsin 12713 Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsinmul 12714 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12707 and cossub 12711. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremcosmul 12715 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12707 and cossub 12711. (Contributed by David A. Wheeler, 26-May-2015.)

Theoremaddcos 12716 Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremsubcos 12717 Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsincossq 12718 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 12719 Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremcos2t 12720 Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcos2tsin 12721 Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)

Theoremsinbnd 12722 The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremcosbnd 12723 The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)

Theoremsinbnd2 12724 The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremcosbnd2 12725 The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremef01bndlem 12726* Lemma for sin01bnd 12727 and cos01bnd 12728. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin01bnd 12727 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 12728 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 12729 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos2bnd 12730 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsinltx 12731 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremsin01gt0 12732 The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremcos01gt0 12733 The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin02gt0 12734 The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos1sgn 12735 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsincos2sgn 12736 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremsin4lt0 12737 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)

Theoremabsefi 12738 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 12739 The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)

Theoremabsefib 12740 A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.)

Theoremefieq1re 12741 A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)

Theoremdemoivre 12742 De Moivre's Formula. Shorter proof of demoivreALT 12743 using the exponential function. (Contributed by NM, 24-Jul-2007.)

TheoremdemoivreALT 12743 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 12744* Lemma for eirr 12745. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeirr 12745 is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremegt2lt3 12746 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 12747 Euler's constant is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)

Theoremepr 12748 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 12749 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 12750 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 11506 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 12751 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 7177 in xpnnen 12749). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremznnenlem 12752 Lemma for znnen 12753. (Contributed by NM, 31-Jul-2004.)

Theoremznnen 12753 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 12754 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 12755* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem2 12756* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremrpnnen2lem3 12757* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem4 12758* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)

Theoremrpnnen2lem5 12759* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem6 12760* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem7 12761* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem8 12762* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem9 12763* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem10 12764* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem11 12765* Lemma for rpnnen2 12766. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2 12766* The other half of rpnnen 12767, 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... 12599). 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 12755). This is an infinite sum of real numbers (rpnnen2lem2 12756), and since implies (rpnnen2lem4 12758) and converges to (rpnnen2lem3 12757) by geoisum1 12597, the sum is convergent to some real (rpnnen2lem5 12759 and rpnnen2lem6 12760) by the comparison test for convergence cvgcmp 12536. The comparison test also tells us that implies (rpnnen2lem7 12761).

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 12762) and cancel them (rpnnen2lem10 12764), 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 12763) and , we establish that (rpnnen2lem11 12765) 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 12767 The cardinality of the continuum is the same as the powerset of . This is a stronger statement than ruc 12783, which only asserts that is uncountable, i.e. has a cardinality larger than . The main proof is in two parts, rpnnen1 10551 and rpnnen2 12766, each showing an injection in one direction, and this last part uses sbth 7177 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 12768 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 7845 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremcpnnen 12769 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)

TheoremrucALT 12770 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 12767, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpeuni/mmcomplex.html#uncountable, see ruc 12783. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremruclem1 12771* Lemma for ruc 12783 (the reals are uncountable). Substitutions for the function . (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

Theoremruclem2 12772* Lemma for ruc 12783. Ordering property for the input to . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem3 12773* Lemma for ruc 12783. The constructed interval always excludes . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem4 12774* Lemma for ruc 12783. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem6 12775* Lemma for ruc 12783. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem7 12776* Lemma for ruc 12783. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem8 12777* Lemma for ruc 12783. The intervals of the sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem9 12778* Lemma for ruc 12783. The first components of the sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem10 12779* Lemma for ruc 12783. 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 12780* Lemma for ruc 12783. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem12 12781* Lemma for ruc 12783. 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 12782 Lemma for ruc 12783. 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 12783 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 12771 through ruclem13 12782 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12782 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 12770. (Contributed by NM, 13-Oct-2004.)

Theoremresdomq 12784 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)

Theoremaleph1re 12785 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)

Theoremaleph1irr 12786 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)

Theoremcnso 12787 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 12788 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.)

Theoremsqr2irr 12789 The square root of 2 is irrational. See zsqrelqelz 13091 for a generalization to all non-square integers. The proof's core is proven in sqr2irrlem 12788, which shows that if , then and are even, so and are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremsqr2re 12790 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)

6.1.2  Some Number sets are chains of proper subsets

Theoremnthruc 12791 The sequence , , , , and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to but not , one-half belongs to but not , the square root of 2 belongs to but not , and finally that the imaginary number belongs to but not . See nthruz 12792 for a further refinement. (Contributed by NM, 12-Jan-2002.)

Theoremnthruz 12792 The sequence , , and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to but not and minus one belongs to but not . This theorem refines the chain of proper subsets nthruc 12791. (Contributed by NM, 9-May-2004.)

6.1.3  The divides relation

Syntaxcdivides 12793 Extend the definition of a class to include the divides relation. See df-dvds 12794.

Definitiondf-dvds 12794* Define the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivides 12795* Define the divides relation. means divides into with no remainder. For example, (ex-dvds 21676). As proven in dvdsval3 12797, . See divides 12795 and dvdsval2 12796 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsval2 12796 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdvdsval3 12797 One nonzero integer divides another integer if and only if the remainder upon division is zero. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)

Theoremdvdszrcl 12798 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremnndivdvds 12799 Strong form of dvdsval2 12796 for natural numbers. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremmoddvds 12800 Two ways to say . (Contributed by Mario Carneiro, 18-Feb-2014.)

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