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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremxpnnenOLD 12801 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 11557 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 12802 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 7219 in xpnnen 12800). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremznnenlem 12803 Lemma for znnen 12804. (Contributed by NM, 31-Jul-2004.)

Theoremznnen 12804 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 12805 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 12806* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem2 12807* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremrpnnen2lem3 12808* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2lem4 12809* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)

Theoremrpnnen2lem5 12810* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem6 12811* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem7 12812* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem8 12813* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem9 12814* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem10 12815* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpnnen2lem11 12816* Lemma for rpnnen2 12817. (Contributed by Mario Carneiro, 13-May-2013.)

Theoremrpnnen2 12817* The other half of rpnnen 12818, 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... 12650). 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 12806). This is an infinite sum of real numbers (rpnnen2lem2 12807), and since implies (rpnnen2lem4 12809) and converges to (rpnnen2lem3 12808) by geoisum1 12648, the sum is convergent to some real (rpnnen2lem5 12810 and rpnnen2lem6 12811) by the comparison test for convergence cvgcmp 12587. The comparison test also tells us that implies (rpnnen2lem7 12812).

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 12813) and cancel them (rpnnen2lem10 12815), 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 12814) and , we establish that (rpnnen2lem11 12816) 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 12818 The cardinality of the continuum is the same as the powerset of . This is a stronger statement than ruc 12834, which only asserts that is uncountable, i.e. has a cardinality larger than . The main proof is in two parts, rpnnen1 10597 and rpnnen2 12817, each showing an injection in one direction, and this last part uses sbth 7219 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 12819 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 7890 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremcpnnen 12820 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)

TheoremrucALT 12821 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 12818, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpeuni/mmcomplex.html#uncountable, see ruc 12834. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremruclem1 12822* Lemma for ruc 12834 (the reals are uncountable). Substitutions for the function . (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)

Theoremruclem2 12823* Lemma for ruc 12834. Ordering property for the input to . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem3 12824* Lemma for ruc 12834. The constructed interval always excludes . (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem4 12825* Lemma for ruc 12834. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem6 12826* Lemma for ruc 12834. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem7 12827* Lemma for ruc 12834. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem8 12828* Lemma for ruc 12834. The intervals of the sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem9 12829* Lemma for ruc 12834. The first components of the sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem10 12830* Lemma for ruc 12834. 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 12831* Lemma for ruc 12834. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremruclem12 12832* Lemma for ruc 12834. 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 12833 Lemma for ruc 12834. There is no function that maps onto . (Use nex 1564 if you want this in the form .) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Theoremruc 12834 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 12822 through ruclem13 12833 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12833 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 12821. (Contributed by NM, 13-Oct-2004.)

Theoremresdomq 12835 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)

Theoremaleph1re 12836 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)

Theoremaleph1irr 12837 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)

Theoremcnso 12838 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 12839 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 12840 The square root of 2 is irrational. See zsqrelqelz 13142 for a generalization to all non-square integers. The proof's core is proven in sqr2irrlem 12839, 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 12841 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 12842 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 12843 for a further refinement. (Contributed by NM, 12-Jan-2002.)

Theoremnthruz 12843 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 12842. (Contributed by NM, 9-May-2004.)

6.1.3  The divides relation

Syntaxcdivides 12844 Extend the definition of a class to include the divides relation. See df-dvds 12845.

Definitiondf-dvds 12845* Define the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdivides 12846* Define the divides relation. means divides into with no remainder. For example, (ex-dvds 21748). As proven in dvdsval3 12848, . See divides 12846 and dvdsval2 12847 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsval2 12847 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdvdsval3 12848 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 12849 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremnndivdvds 12850 Strong form of dvdsval2 12847 for natural numbers. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremmoddvds 12851 Two ways to say . (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdvds0lem 12852 A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds1lem 12853* A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2lem 12854* A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvds 12855 An integer divides itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theorem1dvds 12856 1 divides any integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds0 12857 Any integer divides 0. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegdvdsb 12858 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsnegb 12859 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremabsdvdsb 12860 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsabsb 12861 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Theorem0dvds 12862 Only 0 is divisible by 0 . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul1 12863 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmul2 12864 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremiddvdsexp 12865 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremmuldvds1 12866 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremmuldvds2 12867 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmul 12868 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulc 12869 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdscmulr 12870 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmulcr 12871 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2ln 12872 If an integer divides each of two other integers, it divides any linear combination of them. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2add 12873 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvds2sub 12874 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdstr 12875 The divides relation is transitive. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsmultr1 12876 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremdvdsmultr2 12877 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremordvdsmul 12878 If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremdvdssub2 12879 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsadd 12880 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremdvdsaddr 12881 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssub 12882 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdssubr 12883 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremdvdsadd2b 12884 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremfsumdvds 12885* If every term in a sum is divisible by , then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremdvdslelem 12886 Lemma for dvdsle 12887. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsle 12887 The divisors of a positive integer are bounded by it. The proof does not use . (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsleabs 12888 The divisors of a nonzero integer are bounded by its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theoremdvdseq 12889 If two integers divide each other, they must be equal, up to a difference in sign. (Contributed by Mario Carneiro, 30-May-2014.)

Theoremdvds1 12890 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)

Theoremalzdvds 12891* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremdvdsext 12892* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Theoremfzm1ndvds 12893 No number between and divides . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremfzo0dvdseq 12894 Zero is the only one of the first nonnegative integers that is divisible by . (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremfzocongeq 12895 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^

Theoremdvdsfac 12896 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremdvdsexp 12897 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdvdsmod 12898 Any number whose mod base is divisible by a divisor of the base is also divisible by . This means that primes will also be relatively prime to the base when reduced for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremodd2np1lem 12899* Lemma for odd2np1 12900. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremodd2np1 12900* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

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