mmtheorems81 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10725

Statement List for Metamath Proof Explorer - 8001-8100 - Page 81 of 108

Type	Label	Description
Statement
Theorem	bcthlem8 8001	Lemma for bcth 8027. Any open nonempty set includes a ball of radius less than 1 / (2^k).
Theorem	bcthlem9 8002	Lemma for bcth 8027. If M is rare in X, the intersection of the complement of its closure with any ball is nonempty and open. (Use bcthlem8 8001 for existence of an included ball.)
Theorem	bcthlem10 8003	Lemma for bcth 8027. If M is rare in X, the complement of its closure is not empty and is open.
Theorem	bcthlem11 8004	Lemma for bcth 8027. Triangle inequality.
Theorem	bcthlem12 8005	Lemma for bcth 8027. Helper lemma for satisfying the antecendent of acdc5 7488.
Theorem	bcthlem13 8006	Lemma for bcth 8027. In the sequence g of balls (expressed as ordered pairs), for any m, there is a larger n whose ball's center distance from limit p is less than half of the ball radius at m.
Theorem	bcthlem14 8007	Lemma for bcth 8027. Helper lemma for satisfying the antecendent of acdc5 7488.
Theorem	bcthlem15 8008	Lemma for bcth 8027. Relationship between a ball Q and the next ball P in sequence g, according to the generating function F 's value (KFQ).
Theorem	bcthlem16 8009	Lemma for bcth 8027. A ball in sequence g is included in the complement of the closure of reference sequence M.
Theorem	bcthlem17 8010	Lemma for bcth 8027. The radius of the balls in sequence g decreases exponentially.
Theorem	bcthlem18 8011	Lemma for bcth 8027. Sequence g represents a series of nested balls.
Theorem	bcthlem19 8012	Lemma for bcth 8027. The distance between the center of a ball at m and any later ball in sequence g is less than half the radius of the ball at m.
Theorem	bcthlem20 8013	Lemma for bcth 8027. A weaker version of bcthlem19 8012.
Theorem	bcthlem21 8014	Lemma for bcth 8027. A defining property for (1st o. g) to be a Cauchy sequence.
Theorem	bcthlem22 8015	Lemma for bcth 8027. The sequence of ball centers (1st o. g) is a Cauchy sequence.
Theorem	bcthlem23 8016	Lemma for bcth 8027. Since sequence of ball centers (1st o. g) is a Cauchy sequence and the metric space is complete, the sequence converges to a point p in the metric space.
Theorem	bcthlem24 8017	Lemma for bcth 8027. An upper limit for the distance between a ball center at m and the convergence point q, in terms of any later ball center at n.
Theorem	bcthlem25 8018	Lemma for bcth 8027. Helper lemma to remove the dependence on n of the upper limit in bcthlem24 8017.
Theorem	bcthlem26 8019	Lemma for bcth 8027. The convergence point q belongs to every ball in sequence g.
Theorem	bcthlem27 8020	Lemma for bcth 8027. The convergence point q belongs to the complement of the interior of any member of reference sequence M.
Theorem	bcthlem28 8021	Lemma for bcth 8027. The convergence point q does not belong to any member of reference sequence M.
Theorem	bcthlem29 8022	Lemma for bcth 8027. Therefore the union of all members of reference sequence M does not occupy the entire metric space X. Also, use metric space completeness (via bcthlem23 8016) to eliminate the limit point q from the antecedents.
Theorem	bcthlem30 8023	Lemma for bcth 8027. Apply the Axiom of Dependent Choice acdc5 7488 to show the existence of the recursive sequence of balls g.
Theorem	bcthlem31 8024	Lemma for bcth 8027. Eliminate the antecedents involving sequence g.
Theorem	bcthlem32 8025	Lemma for bcth 8027. Eliminate hypotheses no longer needed.
Theorem	bcthlem33 8026	Lemma for bcth 8027. All members of reference sequence M cannot have an empty interior.
Theorem	bcth 8027	Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, the metric space cannot be the countable union of rare closed subsets (where rare means having an empty interior), so some subset M` k must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.)
|- X = dom dom D   &   |- J = (Open` D)   =>   |- (((D e. CMet /\ X =/= (/) /\ M:NN-->P~X) /\ (U.ran M = X /\ ran M (_ (Clsd` J))) -> E.k e. NN ((int` J)` (M` k)) =/= (/))
Group theory
Definitions and basic properties for groups
Syntax	cgr 8028	Extend class notation with the class of all group operations.
class Grp
Syntax	cgi 8029	Extend class notation with a function mapping a group operation to the group's identity element.
class Id
Syntax	cgn 8030	Extend class notation with a function mapping a group operation to the inverse function for the group.
class inv
Syntax	cgs 8031	Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
class /g
Definition	df-grp 8032	Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54.
|- Grp = {g | E.t(g:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xgy)gz) = (xg(ygz)) /\ E.u e. t A.x e. t ((ugx) = x /\ E.y e. t (ygx) = u))}
Definition	df-gid 8033	Define a function that maps a group operation to the group's identity element.
|- Id = {<.g, y>. | (g e. Grp /\ y = U.{u e. ran g | A.x e. ran g(ugx) = x})}
Definition	df-ginv 8034	Define a function that maps a group operation to the group's inverse function.
|- inv = {<.g, f>. | (g e. Grp /\ f = {<.x, y>. | (x e. ran g /\ y = U.{z e. ran g | (zgx) = (Id` g)})})}
Definition	df-gdiv 8035	Define a function that maps a group operation to the group's division (or subtraction) operation.
|- /g = {<.g, f>. | (g e. Grp /\ f = {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))})}
Theorem	isgrp 8036	The predicate "is a group operation." Note that X is the base set of the group.
|- X = ran G   =>   |- (G e. A -> (G e. Grp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) /\ E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))))
Theorem	isgrpi 8037	Properties that determine a group operation. Read N as N(x).
|- X e. V   &   |- G:(X X. X)-->X   &   |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))   &   |- U e. X   &   |- (x e. X -> (UGx) = x)   &   |- (x e. X -> N e. X)   &   |- (x e. X -> (NGx) = U)   =>   |- G e. Grp
Theorem	grpfo 8038	A group operation maps onto the group's underlying set.
|- X = ran G   =>   |- (G e. Grp -> G:(X X. X)-onto->X)
Theorem	grpcl 8039	Closure law for a group operation.
|- X = ran G   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
Theorem	grplidinv 8040	A group has a left identity element, and every member has a left inverse.
|- X = ran G   =>   |- (G e. Grp -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))
Theorem	grpn0 8041	The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- X = ran G   =>   |- (G e. Grp -> X =/= (/))
Theorem	grpass 8042	A group operation is associative.
|- X = ran G   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
Theorem	grpidinvlem1 8043	Lemma for grpidinv 8047.
Theorem	grpidinvlem2 8044	Lemma for grpidinv 8047.
Theorem	grpidinvlem3 8045	Lemma for grpidinv 8047.
Theorem	grpidinvlem4 8046	Lemma for grpidinv 8047.
Theorem	grpidinv 8047	A group has a left and right identity element, and every member has a left and right inverse.
|- X = ran G   =>   |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
Theorem	grpideu 8048	The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55.
|- X = ran G   =>   |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)
Theorem	grprndm 8049	A group's range in terms of its domain.
|- (G e. Grp -> ran G = dom dom G)
Theorem	0ngrp 8050	The empty set is not a group.
|- -. (/) e. Grp
Theorem	grprn 8051	The range of a group operation. Useful for satisfying group base set hypotheses of the form X = ran G.
|- G e. Grp   &   |- dom G = (X X. X)   =>   |- X = ran G
Theorem	grprnOLD 8052	The range of a group operation. Useful for satisfying X = ran G hypothesis for specific groups.
|- G e. Grp   &   |- G:(X X. X)-->X   =>   |- X = ran G
Theorem	grpidval 8053	The value of the identity element of a group.
|- X = ran G   &   |- U = (Id` G)   =>   |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})
Theorem	grpidcl 8054	The identity element of a group belongs to the group.
|- X = ran G   &   |- U = (Id` G)   =>   |- (G e. Grp -> U e. X)
Theorem	grpidinv2 8055	A group's properties using the explicit identity element.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (((UGA) = A /\ (AGU) = A) /\ E.y e. X ((yGA) = U /\ (AGy) = U)))
Theorem	grplid 8056	The identity element of a group is a left identity.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (UGA) = A)
Theorem	grprid 8057	The identity element of a group is a right identity.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (AGU) = A)
Theorem	grprcan 8058	Right cancellation law for groups.
|- X = ran G   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))
Theorem	grpinveu 8059	The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55.
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> E!y e. X (yGA) = U)
Theorem	grpid 8060	Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- U = (Id` G)   =>   |- ((G e. Grp /\ A e. X) -> (A = U <-> (AGA) = A))
Theorem	grpinvfval 8061	The inverse function of a group.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- (G e. Grp -> N = {<.x, n>. | (x e. X /\ n = U.{y e. X | (yGx) = U})})
Theorem	grpinvval 8062	The inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` A) = U.{y e. X | (yGA) = U})
Theorem	grpinvcl 8063	A group element's inverse is a group element.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` A) e. X)
Theorem	grpinv 8064	The properties of a group element's inverse.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (((N` A)GA) = U /\ (AG(N` A)) = U))
Theorem	grplinv 8065	The left inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> ((N` A)GA) = U)
Theorem	grprinv 8066	The right inverse of a group element.
|- X =