mmtheorems81 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8001-8100 - Page 81 of 107

Type	Label	Description
Statement
Theorem	isgrpi 8001	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 8002	A group operation maps onto the group's underlying set.
|- X = ran G   =>   |- (G e. Grp -> G:(X X. X)-onto->X)
Theorem	grpcl 8003	Closure law for a group operation.
|- X = ran G   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
Theorem	grplidinv 8004	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 8005	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 8006	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 8007	Lemma for grpidinv 8011.
Theorem	grpidinvlem2 8008	Lemma for grpidinv 8011.
Theorem	grpidinvlem3 8009	Lemma for grpidinv 8011.
Theorem	grpidinvlem4 8010	Lemma for grpidinv 8011.
Theorem	grpidinv 8011	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 8012	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 8013	A group's range in terms of its domain.
|- (G e. Grp -> ran G = dom dom G)
Theorem	0ngrp 8014	The empty set is not a group.
|- -. (/) e. Grp
Theorem	grprn 8015	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 8016	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 8017	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 8018	The identity element of a group belongs to the group.
|- X = ran G   &   |- U = (Id` G)   =>   |- (G e. Grp -> U e. X)
Theorem	grpidinv2 8019	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 8020	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 8021	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 8022	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 8023	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 8024	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 8025	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 8026	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 8027	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 8028	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 8029	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 8030	The right inverse of a group element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (AG(N` A)) = U)
Theorem	grpinvid1 8031	The inverse of a group element expressed in terms of the identity element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (AGB) = U))
Theorem	grpinvid2 8032	The inverse of a group element expressed in terms of the identity element.
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> ((N` A) = B <-> (BGA) = U))
Theorem	grpasscan1OLD 8033	An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- U = (Id` G)   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)
Theorem	grpinvid 8034	The inverse of the identity element of a group.
|- U = (Id` G)   &   |- N = (inv` G)   =>   |- (G e. Grp -> (N` U) = U)
Theorem	grplcan 8035	Left cancellation law for groups.
|- X = ran G   =>   |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))
Theorem	isgrp2i 8036	An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57.
|- X e. V   &   |- X =/= (/)   &   |- G:(X X. X)-->X   &   |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))   &   |- ((x e. X /\ y e. X) -> E.z e. X (zGx) = y)   &   |- ((x e. X /\ y e. X) -> E.z e. X (xGz) = y)   =>   |- G e. Grp
Theorem	grpasscan1 8037	An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.)
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (AG((N` A)GB)) = B)
Theorem	grp2inv 8038	Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X) -> (N` (N` A)) = A)
Theorem	grpinvf 8039	Mapping of the inverse function of a group.
|- X = ran G   &   |- N = (inv` G)   =>   |- (G e. Grp -> N:X-1-1-onto->X)
Theorem	grpinvop 8040	The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55.
|- X = ran G   &   |- N = (inv` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (N` (AGB)) = ((N` B)G(N` A)))
Theorem	grpdivfval 8041	Group division (or subtraction) operation.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
Theorem	grpdivval 8042	Group division (or subtraction) operation value.
|- X = ran G   &   |- N = (inv` G)   &   |- D = ( /g ` G)   =>   |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG(N